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![.
Ex. 2: 98 X 97 Base is 100.
ans: 95/06=9506
Ex. 3: 75X95. Base is 100. .
Ex. 4: 986 X 989. Base is 1000
Ex. 5: 994X988. Base is 1000 .
Ex. 6: 750X995
Case 2:
Ex. 7: 13X12. Base is 10
.Ex. 8: 18X14. Base is 10
Ex. 9: 104X102. Base is 100.
104 04
102 02
¯¯¯¯¯¯¯¯¯¯¯¯
Ans is 106 / 4x2 = 10608 ( rule - f ) .
Ex. 10: 1275X1004. Base is 1000.
1275 275
1004 004
...............................................
1279 /275x4 =1279 / 1 / 100 (rule f) =1280100
Case ( iii ): One number is more and the other is less than the base.
Ex.11: 13X7. Base is 10
Ex. 12: 108 X 94. Base is 100.
Ex. 13: 998 X 1025. Base is 1000.
Find the following products by the formula.
1) 7 X 4 2) 93 X 85 3) 875 X 994
4) 1234 X 1002 5) 1003 X 997 6) 11112 X 9998
7) 1234 X 1002 8) 118 X 105
Rules no 2,3,4 of chapter no 1 can also be extended as
Eg. 1: consider 292 x 208. Here 92 + 08 = 100, L.H.S portion
is same i.e. 2
292 x 208 = ( 2 x 3 ) / 92 x 8 / =736 ( for 100 raise the
L.H.S. product by 0 ) = 60736.
Eg. 2: 848 X 852
Here 48 + 52 = 100, L.H.S portion is 8 and its next number is 9.
848 x 852 = 8 x 9 / 48 x 52
720 = 2496
= 722496.
[Since L.H.S product is to be multiplied by 10 and 2 to be carried
over as the base is 100].
Eg. 3: 693 x 607
693 x 607 = 6 x 7 / 93 x 7 = 420 / 651 = 420651.
Find the following products .](https://image.slidesharecdn.com/mathbook-121030080214-phpapp01/75/Math-book-10-2048.jpg)
![divisible by 2 (even) if and only if the whole number is. R 4 - numerals ending
with
a two-digit multiple of 4 represent numbers divisible by 4:o f
Similar to the above, but since 10 is not a multiple of 4, but
10^2 is,
we have to look at the last two digits instead of just the last
digit.
a n 8 - numerals ending with a three-digit multiple of 8
represent
numbers divisible by 8: u Similar to 4, but now 10^2 is not a
multiple
of 8, but 10^3 is, so we have to look at the last three mdigits.
er
a
16 - You figure this one!
l
5 - You figure this one, too!
ook
3 - numerals whose sum of digits is divisible by 3 represent
numbers
s divisible by 3: This one is different, because 3 does not
divide any
power of 10 evenly. That means we will have to consider the
effect of
all the l digits. Here we use this fact: 10^k - 1 = (10 - 1)*(10^(k-
1) = ... +
10^2 + i 10 + 1) This is a fancy way of saying 9999...999 =
9*1111...111
. We use k this to rewrite our powers of 10 as 10^k = 9*a[k] +
1. Now the
polynomial e this: d[k]*10^k + d[k-1]*10^(k-1) + ... + d[1]*10 +
d[0] = d[k
]*(9*a[k] + 1) + ... + d[1]*(9*a[1] + 1) + d[0] = 9*(d[k]*a[k] + ... +
d[1]*a[1])](https://image.slidesharecdn.com/mathbook-121030080214-phpapp01/75/Math-book-41-2048.jpg)
![+ d[k] + ... + d[1] + d[0] Now notice that 3 divides the first
part, so the
whole number is divisible by 3 if and only if the sum of the digits is.
9 - numerals whose sum of digits is divisible by 9 represent
numbers
divisible by 9: Use the same equation as the previous case.
You figure
the rest!
11 - numerals whose alternating sum of digits is divisible by
11 represent
numbers divisible by 11:
Here the phrase "alternating sum" means we alternate the
signs from
positive to negative to positive to negative, and so on. We
use this fact:
10 to an odd power plus 1 is divisible by 11, and 10 to an
even power
minus 1 is divisible by 11. The first part is a fancy way of
writing 10000.
..0001 = 11*9090...9091 (where there are an even number of
0's on the
left-hand side). The second part is a fancy way of writing
99999...9999
= 11*9090...0909 (where there are an even number of 9's in
the left
hand side). We write 10^(2*k) = 11*b[2*k] + 1 and 10^(2*k+1) =
11*b
[2*k+1] - 1. Here k is any nonnegative integer. We substitute
that
into the polynomial form, so:d[2*k]*10^(2*k) + d[2*k-
1]*10^(2*k-1) + .
.. + d[1]*10 + d[0]
= d[2*k]*(11*b[2*k] + 1) + d[2*k-1]*(11*b[2*k-1] - 1) + ...](https://image.slidesharecdn.com/mathbook-121030080214-phpapp01/75/Math-book-42-2048.jpg)
![� + d[1]*(11*a[1] - 1) + d[0] = 11*(d[2*k]*b[2*k] + ... + d[1]*a[1])
+ d[2*k]
- d[2*k-1]
� + ... - d[1] + d[0]
The first part is divisible by 11 no matter what the digits are,
so the
whole number is divisible by 11 if and only if the last part,
which is
the alternating sum of the digits, is
divisible by 11. If you prefer, you can write
d[2*k] - d[2*k-1] + ... - d[1] + d[0] = (d[0] + d[2] + ... + d[2*k]) -
(d[1] +
d[3] + ... + d[2*k-1]), so that you add up every other digit,
starting from
the units digit, and then add up the remaining digits, and
subtract the
two sums. This will compute the same result as the
alternating sum of
the digits. 7 - There is a trick for 7 which is not as well known
as the
others. It makes use of the fact that 10^(6*k) - 1 is divisible by
7, and
10^(6*k - 3) + 1 is divisible by 7. It goes like this: Mark off the
digits in
groups of threes, just as you do when you put commas in
large numbers.
Starting from the right, compute the alternating sum of the
groups as
three-digit numbers. If the result is negative, ignore the sign.
If the
result is greater than 1000, do the same thing to the resulting
number
until you have a result between 0 and 1000 inclusive. That 3-
digit](https://image.slidesharecdn.com/mathbook-121030080214-phpapp01/75/Math-book-43-2048.jpg)
![if it's simple enough, but it's the only explanation I can think
of.
Doctor Darren, The Math Forum
Check out our web site -http://forum.swarthmore.edu/dr.math/
Dividing by 4
ook at the last two digits. If they are divisible by 4, the
number is
as well. Examples:
1 100 is divisible by 4. 2 1732782989264864826421834612 is
divisible by four also, because 12 is divisible by four.
Dividing by 5
If the last digit is a five or a zero, then the number is divisible
by 5.
Dividing by 6
Check 3 and 2. If the number is divisible by both 3 and 2, it is
divisible by 6 as well. Robert Rusher writes in:
Another easy way to tell if a [multi-digit] number is
divisible by six . . .
is to look at its [ones digit]: if it is even, and the sum of
the [digits] is
a multiple of 3, then the number is divisible by 6.
Dividing by 7
To find out if a number is divisible by seven, take the last
digit,
double it, and subtract it from the rest of the number.
Example:
If you had 203, you would double the last digit to get six, and
ubtract that from 20 to get 14. If you get an answer divisible
by 7
(including zero), then the original number is divisible by
seven.
If you don't know the new number's divisibility, you can
apply
the rule again.
Dividing by 8](https://image.slidesharecdn.com/mathbook-121030080214-phpapp01/75/Math-book-64-2048.jpg)
![application of modulo arithmetic leads to divisibility rules
that can
be used. These rules are important to mathematics because
they
save us lots of time and effort.
In fact, many of the divisibility rules that are commonly used
(rules for 3, 9, 11) have their roots in modulo arithmetic. This
page will show you how some of these common divisibility
rules are connected to modulo arithmetic.
Applying the Rules of Modulo Arithmetic
The rules of modulo arithmetic state that the number N is
divisible
by some number (P) if the above expression is also divisible
by P
after the base (10) is replaced by the remainder of 10 divided
by P.
In compact notation, this remainder is denoted by (10 mod
P). T
hus, N is divisible by P if the following expression is also
divisible by P:
[Dn * (10 mod P)^(n-1)] + ... + [D2 * (10 mod P)^1] + [D1 * (10
mod P)^0].
Origin of Divisibility Rules for 3, 9, and 11
D We can now see how the divisibility rules for P = 3, 9, and
11 are
rooted in modulo arithmetic. n
Consider the case where P = 3. .
Because 10 mod 3 is equal to 1, any number is divisible by 3
if the
following expression is
. also divisible by 3: .
-
[Dn * (1)^n-1] + ... + [D2 * (1)^1] + [D1* D
2
(1)^0]. Because 1 raised to any power is equal to 1, the above](https://image.slidesharecdn.com/mathbook-121030080214-phpapp01/75/Math-book-70-2048.jpg)
![+
expression can be simplified as: Dn + ... + D2 + D1 + D0,
D
which is equal to the sum of digits in the number. Thus
modulo
arithmetic allows us to 1
state that any number is divisible by 3 if the sum of its digits is
Because 10 mod 9 is also 1, any number is divisible by 9 is
the
sum of its digits are also
divisible by 9. Thus, the divisibility rules for 3 and 9 come
directly
from modulo n
arithmetic.
n)
Finally, consider the case where P = 11. a
Because 10 mod 11 is equal to -1, any number is divisible by
11
if the n following expression is also divisible by 11: d
D[Dn * (-1)^n-1] + ... + [D2 * (-1)^1] + [D1* (-1)^0].
n
Because -1 raised to any even power is equal to 1 and -1
raised to
any odd +power is equal to -1, the above expression becomes
a summation of the digits involving alternated signs: .
. . - D2 + D1 (for odd n).
This "alternating" summation rule for P = 11 is well known
and has two versions. The first version states a number is
divisible by 11 if the alternating sum of the digits is also
divisilbe
by 11 (i.e., the alternating sum is a muliple of 11). The second
version states that a number is divisible by 11 if the sum of
every
other digit starting with the rightmost digit is equal to the
sum of](https://image.slidesharecdn.com/mathbook-121030080214-phpapp01/75/Math-book-71-2048.jpg)
![every other digit starting with the second digit from the right
(i.e.,
the alternating sum is 0). Both of these versions come
directly
from the rules of modulo arithmetic.
Bases Other Than 10
v Up to this point, we have only considered numbers written
in
base 10. A number can, however, be i written in any base (B).
Such a number can be expressed by the following
summation: g
it N = [Dn * B^n-1] + [Dn-1 * B^(n-2)] + ... + [D2 * B^1] + [D1*
B^0]. s
"
The rules of modulo arithmetic apply no matter what base is
used
. These rules tell us that N is divisible by P if the following
expression is also divisible by P: [Dn * (B mod P)^n-1] + ...
+ [D2 * (B mod P)^1] + [D1* (B mod P)^0]. Most bases other
than 10 are difficult to use, but bases
that are powers of 10 can easily be
used to construct divisibility rules using modulo arithmetic.
Consider writing the base-10 number N = 1233457 in base-
100.
This can be done by starting with the rightmost digit and
grouping the digits in pairs of two. bEach grouping of
two digits is considered a single "digit" when the number
is ywritten in base-100. Written in base-100, the number is
1 23 34 57 where 57, 34, 123, and 1 are considered single
"digits." In a similar fashion, this number written 1in
base-1000 is 1 233 457 where 457, 233, and 1 are considered
single "digits." .Once the base-10 digits are grouped to form
the "digits," the above expression Bcan be used to test for
divisibility. Consider P = 7.
Use the base, B = 1000. Because 1000 mod 7 = -1, the](https://image.slidesharecdn.com/mathbook-121030080214-phpapp01/75/Math-book-72-2048.jpg)
Uploaded byDinesh Lambat
Math book
This document provides techniques for quickly solving multiplication and square root problems for bank exams using tricks and shortcuts. It explains methods for multiplying and finding squares of two-digit numbers when the sum of the last digits is 10 or 5. Rules and ranges are given for finding squares of any number based on their proximity to base numbers like 50, 100, 150 etc. Steps are outlined for separating a number into units and tens/hundreds and determining its square root based on ranges of perfect squares. Examples are included to demonstrate each method.
Math book
- 1. Hi, This book isbased on some quicker methods for different numerical problems, hence useful for Bank Exams. MULTIPLICATION TABELE (10x10) 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 4 6 8 10 12 14 16 18 20 3 9 12 15 18 21 24 27 30 4 16 20 24 28 32 36 40 5 25 30 35 40 45 50 6 Think As 8*4 4 *8,.. 36 42 48 54 60 7 49 56 63 70 8 68 72 80 9 81 90 10 100 MULTIPLICATION TABLE (25x10) 21 21 42 63 84 105 126 147 168 189 210 22 22 44 66 88 110 132 154 176 198 220 23 23 46 69 92 115 138 161 184 207 230 24 24 48 72 96 120 144 168 192 216 240 25 25 50 75 100 125 150 175 200 225 250 1 2 3 4 5 6 7 8 9 10 11 11 22 33 44 55 66 77 88 99 110 12 12 24 36 48 60 72 84 96 108 120 13 13 26 39 52 65 78 91 104 117 130 14 14 28 42 56 70 84 98 112 126 140 15 15 30 45 60 75 90 105 120 135 150 16 16 32 48 64 80 96 112 128 144 160 17 17 34 51 68 85 102 119 136 153 170 18 18 36 54 72 90 108 126 144 162 180 19 19 38 57 76 95 114 133 152 171 190 20 20 40 60 80 100 120 140 160 180 200 MULTIPLICATION TABLE (11-20 x11-20) 11 12 13 14 15 16 17 18 19 20
- 2. 11 121 132143 154 165 176 187 198 209 220 12 144 156 168 180 192 204 216 228 240 13 169 182 195 208 221 234 247 260 14 196 210 224 238 252 266 280 15 225 240 255 270 285 300 16 256 272 288 304 320 17 289 306 323 340 18 324 342 360 19 361 380 20 400 For quickest math- 1. Many diagrams are given in almost every example which would be the reminder of the process when u completely learn the techniques. 2. Try to understand the colors, every color is explaining the steps and other important things. Edited by RAKES PRASAD First we learn how to find square, square root and cube roots with multiplication techniques. Rule no 1. You are to remember squares of 1 to 32. They are given as follows No Square No Sq No Sq No Sq no sq 1 1 9 81 17 289 25 625 33 1089 2 4 10 100 18 324 26 676 34 1156 3 9 11 121 19 361 27 729 35 1225 4 16 12 144 20 400 28 784 36 1296 5 25 13 169 21 441 29 841 38 1444 6 36 14 196 22 484 30 900 41 1681 7 49 15 225 23 529 31 961 8 64 16 256 24 576 32 1024 N.B. The square of 38 contains 444 and the sq of an odd prime no 41 contains two perfect squares as 4 2 = 16 and 9 2 = 81. Rule No 2. Squares of numbers ending in 5 :we are showing by giving an example. For the number 25, the last digit is 5 and the 'previous' digit is 2. 'to multiply the previous digit 2 by one more , that is, by 3'. It becomes the L.H.S of the result, that is, 2 X 3 = 6. The R.H.S of the result is
- 3. 5 2 , that is,25. Thus 25 2 = 2 X 3 / 25 = 625. In the same way, 105 2 = 10 X 11/25 = 11025; 135 2 = 13 X 14/25 = 18225; see the figure below… …. Now we are extending the formula as Rule no 3: If sum of the last two digits give 10 then you can use the formula but L.H.S should be same. Check the examples Ex 1 : 47 X 43 .See the end digits sum 7 + 3 = 10 ; then 47 x 43 = ( 4 + 1 ) x 4 / 7 x 3 = 20 / 21 = 2021. Example 2: 62 x 68 2 + 8 = 10, L.H.S. portion remains the same i.e.,, 6. Next of 6 gives 7 and 62 x 68 = ( 6 x 7 ) / ( 2 x 8 ) = 42 / 16 = 4216. Example 3: 127 x 123 127 x 123 = 12 x 13 / 7 x 3 = 156 / 21 = 15621. (see the fig ) Example 4. 395 2 395 2 = 395 x 395 = 39 x 40 / 5 x 5 = 1560 / 25 = 156025. You can rem the formula as 125 2 =(12 2 +12)25=(144+12)25=15625. Both are same things but it can use in bigger case easily. Rule 4: Also we can extend the formula where the sum of last two digits being 5 as Ex 1. 82 x 83=(8 2 +8/2) /3 x 2=(64+4) / 06=6806 i.e.( n 2 +n/2) n is even but note that the R.H.S term should be of two digits as 06 .( in this ex.)
- 4. Ex 2.181 x184=(18 2 +9) / 04=(324+9) / 04=33304. But we know what are u thinking of? Yes, if the number be odd then what happen? And ur ans is as follows: Ex 3: 91 x 94= (9 2 +9/2) / 04=(81+4) / 54. Surprised? We just pass ½ as 5 in the next row since ½ belongs to hundred’s place so 1/2 x 100=50. Just pass 5 in such cases. See the fig. Ex 4 . 51 x 54= (25+2) / 54 =2754 Ex 5. 171 x 174= (289+8) / 54=29754 Rule no 5 . Now we are discussing the general method of finding squares of any number. See the chart first. See 5 2 =25 And 10 2 = 100 , so on. Now u can understand the tricks. base range Trick alternative 50 26-74 25(±)( read as 25 plus minus) 100 76-126 100(±)2(±) Given no(±) 150 126-174 225(±)3(±) 200 176-224 400(±)4(±) 250 226-274 625(±)5(±) 300 276-224 900(±)6(±) 350 326-374 1225(±)7(±) 400 376-424 1600(±)8(±) 450 426-474 2025(±)9(±) 500 476-524 2500(±)10(±) ……… ………….. ………………………. 1000 976-1024 100,00(±)20(±) 1500 1476-1524 225,00,00(±)30(±) 2000 1976-2024 4,000,000(±)40(±) For base 50: Now we are explaining one by one through some examples. Ex 1. 43 2
- 5. step 1. Findbase –no as 50-43=07 Step 2. Sq the no as 07 2 =49.write it on the right side. Step 3. Subtract the no from 25 as 25-7=18.write it on extreme left. Step 4. Now this is Ur ans. 1849 Ex 2. 57 2 step 1. 57-50=7 Step2. 7 2 =49 Step3. 25+7=32 step4. Ans is 3249. Ex 3. 69 2 step1. 69-60=19 , 19 2 =361 Step2. carry over 3 and write 61 in right side. Step3 . 25+19=44, now add 44 with 3 (carry number) and write it with extreme left. i.e. (25+19+3) Step 4. ur ans is 4761 Ex4 . 38 2 step1. 50-38=12 , 12 2 =144 ,carry 1 and write 44 at right side . Step2. now (25-12)+1=14 i.e. Carry number always adds (it never subtract). So ur ans is 1444. So, 1. At first see the given no is it more or less than Ur base. If it is more then addition rule and if it be less, the n subtraction rule will be applied. That’s why (±) sign is given 2. And always add the carried no. 3. range is chosen making (±) 24 from base . 4. Trick is done by squaring the base excluding a 0 (zero) such for base 50 : 5 2 =25.for base 350 , 35 2 = 1225 5. for(±) divide the base without zero by 5 as for base 100: 10/5=2,so trick is 100 (±)2(±) and for base 350: 35/5=7 ,and the trick is 1225(±)7(±) and so on.
- 6. Now we aregiving more examples to clear the fact, a regular practice can make u master of the art.ok, see The following examples. For base 100 Ex 5. 89 2 Step1. 100-89=11 ,11 2 =121 ,carry 1 Step2. 100-(2 x 11 )+1=79, so the ans is 7921 Also u can use the alternative formula here,”given no (±)” such as Step 1. 100-89=11,sq is 121,carry 1 Step2. 89-11+1=79 ,ass is 7921 (I think this process is better for numbers whose base is near 100) Ex 6. 117² Step1. 117-17=17, 17 2 =289 ,carry 2 Step2. 117+17=134, 134+2=136 ,ans is 13689 For base 150 Ex 6. 139² 1. 150-139=11 , sq is 121 ,carry 1 2. 225-3×11=225-33=192, 192+1=193 ,ans is 19, 321 Ex 7. 164 ² 1. 164-150=14, 14 ² =196, carry 1 2. 225+3(14)=225+42=267 ,267+1=268 ,ans is 26,896 Ex 8. 512 ² 1. 12 ² =144 ,carry 1, 2500+10(12)+1=2621, ans is 262144 Try yourself to various numbers to learn the technique quickly. now we learn to find the square root .before starting note that… Rule no 6: unit digit of perfect square 1 4569 unit digit of square root 1,9 (1+9)=10 2,8 2+8=10 5 5+5=10 4,6 4+6=10
- 7. 3,7 3+7=10 Notes : 1. any no end with 2,3,7,8 can’t be a perfect square. 2. any no ends with odd no zero’s can’t be a perfect square. Now check the given boxes… range sq range sq range sq range 1²=1 1-3 9²=81 81-99 17²=289 289-323 25²=625 625-675 2²=4 4-8 10²=100 100-120 18²=324 324-360 26²=676 676-728 3²=9 9-15 11²=121 121-143 19²=361 361-399 27²=729 729-783 4²=16 16-24 12²=144 144-168 20²=400 400-440 28²=784 784-840 5²=25 25-35 13²=169 169-195 21²=441 441-483 29²=841 841-899 6²=36 36-48 14²=196 196-224 22²=484 484-528 30²=900 900-960 7²=49 49-63 15²=225 225-255 23²=529 529-575 31²=961 961- 1023 8²=64 64-80 16²=256 256-288 24²=576 576-624 32²=1024 1024- 1088 Rule: 1. separate the two numbers of extreme right. 2. find the range of the square of the rest, write it on left side. 3. Multiply the range with its preceding number and see that the separated number is more or less than the multiplied number. 4. if more, then choose the large no of unit digit as the R.H.S. Ex1. √ (2601) 1. 26 / 01 2. 26 falls on the range of 5 ,write 5 in the left. 3. now 5×6=30 ,26 is smaller than 30. 4. so choose 1 as the right side no. ans is 51 Ex2. √ (6241) 1. 62 / 41 2. 62 falls in range of 7 ,write 7 in left. 3. 7×8=56 i.e.,62 is more than 56,so we choose 9 as the right side Ex3. √ 2704 1. 27 / 04 2. 27 falls in range of 5, so 5×6=30 ,but 27 is less than 30, so We choose 2 between 2 & 8 (allocated for 4) 3.so the ans is 52 But it is easy if the number ends with 5 such as Ex 4. √ (99225) 1. 992 /25 2. Here u can insert 5 as right side because there is only 5 allocated for 5 and no choice need here.
- 8. 3. clearly 992falls in the range of 31 and the ans is 315 Exercise 1. √ 34225 2. √ 105625 3. √ (0.00126025) 4. √ 2209 5. √ ___________________2916 6. √ 2116 7. √ 15129 8. √ 16129 9. √ 55696 10. √ 66564 11. √ 8.8804 12. √ 0.00101124 13. √ 0.125316 Rule no 7: let us know how to find cube root quickly. See the box first. unit digit of perfect cube 1 2 3 4 5 6 7 8 9 unit digit of cube root 1 8 2+8=10 7 3+7=10 4 5 6 3 7+3=10 2 8+2=10 9 n.b. 1, 4 , 5, 6, 9 has no change as they were in the sq formula and 2,3,7,8 has there compliment with 10, i.e., 2+8=10, 3+7=10 etc. Ex1. (46656) 1/3 1. 46 / 656 2. write 6 for 6 on right side 3. see 46 falls in the range of 3 3 4.so the ans is 36 cube range cube range cube range 1 =1 1-7 6 =216 216-342 11 =1331 1331-1727 2 =8 8-26 7 =343 343-511 12 =1728 1728-2196 3 =27 27-63 8 =512 512-728 13 =2197 2197-2743 4 =64 64-124 9 =729 729-999 14 =2744 2744-3374 5 =125 125-215 10 =1000 1000-1330 15 =3375 3375-4095 21=9261 25=15625 Ex2 . cube of 0.0020048383 1. 0.0020048 / 383 2. write 7 for 3 , and see 2048 falls in the range of 12
- 9. 3. so thens be 0 .123 n.b. (se 3 places to shift 1 decemal place. ) Rule no 8: Let us know to find the cube of any two digits number. Ex 1. (18) 3 Step1. Find the ratio of the numbers such as 1:8 in this example. Step 2. Write cube of first digit and then write three successive terms in horizontal line which are multiplied by the ratio. in this ex as Step 3. Make double of second and third terms and write them just below their own positions Step 4. Add successive terms (carry over if more than one digit) and u will get the required result .Lets see the methods … 1. 18 , that is 1:8 4 24 51 2. 1 3 =1 / 1x8=8 / 8 x 8=64 / 64 x 8 =512 3. 8 x 2=16 64 x 2=128 4. ------------------------------------------------------------ ( 4 +1) ( 24 +8+16) ( 51 +64+128) 51 2 5 = 4 8 = 24 3 5. So the ans is 5832. Ex 2. (33) 3 Rakes Prasad , 2008 It contains some vedic methds for multiplication of two,three and more digits,finding h.c.f. ofr two polynomial equations. Edited by Rakes Prasad, 2008 Rule no (1) If R.H.S. contains less number of digits than the number of zeros in the base, the remaining digits are filled up by giving zero or zeroes on the left side of the R. Note: If the number of digits are more than the number of zeroes in he base, the excess digit or digits are to be added to L.H.S of the answer. Case 1: Let N1 and N2 be two numbers near to a given base in powers of 10, and D1 and D2 are their respective deviations(difference) from the base. Then N1 X N2 can be represented as Ex. 1: Find 97 X 94. Here base is 100.
- 10. . Ex. 2: 98X 97 Base is 100. ans: 95/06=9506 Ex. 3: 75X95. Base is 100. . Ex. 4: 986 X 989. Base is 1000 Ex. 5: 994X988. Base is 1000 . Ex. 6: 750X995 Case 2: Ex. 7: 13X12. Base is 10 .Ex. 8: 18X14. Base is 10 Ex. 9: 104X102. Base is 100. 104 04 102 02 ¯¯¯¯¯¯¯¯¯¯¯¯ Ans is 106 / 4x2 = 10608 ( rule - f ) . Ex. 10: 1275X1004. Base is 1000. 1275 275 1004 004 ............................................... 1279 /275x4 =1279 / 1 / 100 (rule f) =1280100 Case ( iii ): One number is more and the other is less than the base. Ex.11: 13X7. Base is 10 Ex. 12: 108 X 94. Base is 100. Ex. 13: 998 X 1025. Base is 1000. Find the following products by the formula. 1) 7 X 4 2) 93 X 85 3) 875 X 994 4) 1234 X 1002 5) 1003 X 997 6) 11112 X 9998 7) 1234 X 1002 8) 118 X 105 Rules no 2,3,4 of chapter no 1 can also be extended as Eg. 1: consider 292 x 208. Here 92 + 08 = 100, L.H.S portion is same i.e. 2 292 x 208 = ( 2 x 3 ) / 92 x 8 / =736 ( for 100 raise the L.H.S. product by 0 ) = 60736. Eg. 2: 848 X 852 Here 48 + 52 = 100, L.H.S portion is 8 and its next number is 9. 848 x 852 = 8 x 9 / 48 x 52 720 = 2496 = 722496. [Since L.H.S product is to be multiplied by 10 and 2 to be carried over as the base is 100]. Eg. 3: 693 x 607 693 x 607 = 6 x 7 / 93 x 7 = 420 / 651 = 420651. Find the following products .
- 11. 1. 318 x312 2. 425 x 475 3. 796 x 744 4. 902 x 998 5. 397 x 393 6. 551 x 549 (2) 'One less than the previous' 1) The use of this sutra in case of multiplication by 9,99,999.. is as follows a) The left hand side digit (digits) is ( are) obtained by deduction 1 from the left side digit (digits) . b) The right hand side digit is the complement or difference between the multiplier and the left hand side digit (digits) c) The two numbers give the answer Example 1: 8 x 9 Step ( a ) gives 8 – 1 = 7 ( L.H.S. Digit ) Step ( b ) gives 9 – 7 = 2 ( R.H.S. Digit ) Step ( c ) gives the answer 72 Example 2: 15 x 99 Step ( a ) : 15 – 1 = 14 Step ( b ) : 99 – 14 = 85 ( or 100 – 15 ) Step ( c ) : 15 x 99 = 1485 Example 3: 24 x 99 Answer : Example 5: 878 x 9999 Answer : find out the products 64 x 99 723 x 999 3251 x 9999 43 x 999 256 x 9999 1857 x 99999 (3) Multiplication of two 2 digit numbers. Ex.1: Find the product 14 X 12 The symbols are operated from right to left . Step i) : Step ii) : Step iii) : Ex.4: 32 X 24 Step (i) : 2 X 4 = 8 Step (ii) : 3 X 4 = 12; 2 X 2 = 4; 12 + 4 = 16. Here 6 is to be retained. 1 is to be carried out to left side. Step (iii) : 3 X 2 = 6. Now the carried over digit 1 of 16 is to be added. i.e., 6 + 1 = 7. Thus 32 X 24 = 768 Note that the carried over digit from the result (3X4) + (2X2) = 12+4 = 16 i.e., 1 is placed under the previous digit 3 X 2 = 6 and added. After sufficient practice, you feel no necessity of writing in this way and simply operate or perform mentally. (4) Consider the multiplication of two 3 digit numbers.
- 12. Ex 1. 124X 132 =16368. Proceeding from right to left i) 4 X 2 = 8. First digit = 8 ii) (2 X 2) + (3 X 4) = 4 + 12 = 16. The digit 6 is retained and 1 is carried over to left side. Second digit = 6. iii) (1 X 2) + (2 X 3) + (1 X 4) = 2 + 6 + 4 =12. The carried over 1 of above step is added i.e., 12 + 1 = 13. Now 3 is retained and 1 is carried over to left side. Thus third digit = 3. iv) ( 1X 3 ) + ( 2 X 1 ) = 3 + 2 = 5. the carried over 1 of above step is added i.e., 5 + 1 = 6 . It is retained. Thus fourth digit = 6 v) ( 1 X 1 ) = 1. As there is no carried over number from the previous step it is retained. Thus fifth digit = 1 (5) Cubing of Numbers: Example : Find the cube of the number 106. We proceed as follows: i) For 106, Base is 100. The surplus is 6. Here we add double of the surplus i.e. 106+12 = 118. (Recall in squaring, we directly add the surplus) This makes the left-hand -most part of the answer. i.e. answer proceeds like 118 / - - - - - ii) Put down the new surplus i.e. 118-100=18 multiplied by the initial surplus i.e. 6=108. Since base is 100, we write 108 in carried over form 108 i.e. . As this is middle portion of the answer, the answer proceeds like 118 / 108 /.... iii) Write down the cube of initial surplus i.e. 63 = 216 as the last portion i.e. right hand side last portion of the answer. Since base is 100, write 216 as 216 as 2 is to be carried over. Answer is 118 / 108 / 216 Now proceeding from right to left and adjusting the carried over, we get the answer 119 / 10 / 16 = 1191016. Eg.(1): 1023 = (102 + 4) / 6 X 2 / 23 = 106/ 12 / 08 = 1061208. Observe initial surplus = 2, next surplus =6 and base = 100. Eg.(2): 943 Observe that the nearest base = 100. i) Deficit = -6. Twice of it -6 X 2 = -12 add it to the number = 94 -12 =82. ii) New deficit is -18. Product of new deficit x initial deficit = -18 x -6 = 108 iii) deficit3 = (-6)3 = -216. __ Hence the answer is 82 / 108 / -216 Since 100 is base 1 and -2 are the carried over. Adjusting the carried over
- 13. in order, weget the answer ( 82 + 1 ) / ( 08 – 03 ) / ( 100 – 16 ) = 83 / 05 / 84 = 830584 . 16 becomes 84 after taking1 from middle most portion i.e. 100. (100-16=84). _ Now 08 - 01 = 07 remains in the middle portion, and 2 or 2 carried to it makes the middle as 07 - 02 = 05. Thus we get the above result. Eg.(3): 9983 Base = 1000; initial deficit = - 2. 9983 = (998 – 2 x 2) / (- 6 x – 2) / (- 2)3 = 994 / 012 / -008 = 994 / 011 / 1000 - 008 = 994 / 011 / 992 = 994011992. Find the cubes of the following numbers using yavadunam sutra. 1. 105 2. 114 3. 1003 4. 10007 5. 92 6. 96 7. 993 8. 9991 9. 1000008 10. 999992. (6) Highest common factor: Example 1: Find the H.C.F. of x2 + 5x + 4 and x2 + 7x + 6. 1. Factorization method: x2 + 5x + 4 = (x + 4) (x + 1) x2 + 7x + 6 = (x + 6) (x + 1) H.C.F. is ( x + 1 ). 2. Continuous division process. x2 + 5x + 4 ) x2 + 7x + 6 ( 1 x2 + 5x + 4 ___________ 2x + 2 ) x2 + 5x + 4 ( ½x x2 + x __________ 4x + 4 ) 2x + 2 ( ½ 2x + 2 ______ 0 Thus 4x + 4 i.e., ( x + 1 ) is H.C.F. OUR PROCESS i.e.,, (x + 1) is H.C.F Example 2: Find H.C.F. of 2x2 – x – 3 and 2x2 + x – 6 Example 3: x3 – 7x – 6 and x3 + 8x2 + 17x + 10. Example 4: x3 + 6x2 + 5x – 12 and x3 + 8x2 + 19x + 12. (or) Example 5: 2x3 + x2 – 9 and x4 + 2x2 + 9 Add: (2x3 + x2 – 9) + (x4 + 2x2 + 9) = x4 + 2x3 + 3x2. ÷ x2 gives x2 + 2x + 3 ------ (i) Subtract after multiplying the first by x and the second by 2. Thus (2x4 + x3 – 9x) - (2x4 + 4x2 + 18) = x3 - 4x2 – 9x – 18 ------ ( ii ) Multiply (i) by x and subtract from (ii) x3 – 4x2 – 9x – 18 – (x3 + 2x2 + 3x) = - 6x2 – 12x – 18
- 14. ÷ - 6gives x2 + 2x + 3. Thus ( x2 + 2x + 3 ) is the H.C.F. of the given expressions. Find the H.C.F. in each of the following cases using Vedic sutras: 1 x2 + 2x – 8, x2 – 6x + 8 2 x3 – 3x2 – 4x + 12, x3 – 7x2 + 16x - 12 3 x3 + 6x2 + 11x + 6, x3 – x2 - 10x - 8 4 6x4 – 11x3 + 16x2 – 22x + 8, 6x4 – 11x3 – 8x2 + 22x – 8. Edited by Rakes Prasad , 2008 Contains: (1) Use the formula ALL FROM 9 AND THE LAST FROM 10 to perform instant subtractions. (2) Multiplying numbers just over 100. (3) The easy way to add and subtract fractions. (4) Multiplying a number by 11. (5) Method for diving (6) SOME BASIC RELATIONS AND REVIEW OF SQUARE AND CUBE FORMULA: (7) SQUARE ROOTS AND MULTIPLICATION FORMULA : (8) TWO INTERESTING FACTS: (9) ROMAN NUMERALS (10) NUMER REPRESENTATION: (11) DECIMAL REPRENTATION AND PREFIXS: (12) POWER OF INTIGERS: (13) SOME PROOFS BUT WITHOUT WORDS: (14) THE LARGEST PRIME NUMBERS DISCOVERED SO FAR: Edited by Rakes Prasad, 2008 (1) Use the formula ALL FROM 9 AND THE LAST FROM 10 to perform instant subtractions. ● For example 1000 - 357 = 643 We simply take each figure in 357 from 9 and the last figure from 10. So the answer is 1000 - 357 = 643 And thats all there is to it! This always works for subtractions from numbers consisting of a 1 followed by noughts: 100; 1000; 10,000 etc.
- 15. Similarly 10,000 -1049 = 8951 For 1000 - 83, in which we have more zeros than figures in the numbers being subtracted, we simply suppose 83 is 083. So 1000 - 83 becomes 1000 - 083 = 917 Try some yourself: 1) 1000 - 777 2) 1000 - 283 3) 1000 - 505 4) 10,000 - 2345 5) 10000 - 9876 6) 10,000 - 1101 (2) Multiplying numbers just over 100. ● 103 x 104 = 10712 The answer is in two parts: 107 and 12, 107 is just 103 + 4 (or 104 + 3), and 12 is just 3 x 4. ● Similarly 107 x 106 = 11342 107 + 6 = 113 and 7 x 6 = 42 Again, just for mental arithmetic Try a few: 1) 102 x 107 = 2) 106 x 103 = 1) 104 x 104 = 4) 109 x 108 = (3) The easy way to add and subtract fractions. Use VERTICALLY AND CROSSWISE to write the answer straight down! Multiply crosswise and add to get the top of the answer: 2 x 5 = 10 and 1 x 3 = 3. Then 10 + 3 = 13. The bottom of the fraction is just 3 x 5 = 15. . You multiply the bottom number together. Subtracting is just as easy: multiply crosswise as before, but the subtract: Try a few: (4) Multiplying a number by 11. To multiply any 2-figure number by 11 we just put the total of the two figures between the 2 figures. ● 26 x 11 = 286 Notice that the outer figures in 286 are the 26 being multiplied. And the middle figure is just 2 and 6 added up. ● So 72 x 11 = 792 Multiply by 11: 1) 43 = ) 81 = 3) 15 = 4) 44 = 5) 11 = ● 77 x 11 = 847 This involves a carry figure because 7 + 7 = 14 we get 77 x 11 = 7147 = 847. Multiply by 11: 1) 88 = 2) 84 = 3) 48 = 4) 73 = 5) 56 = ● 234 x 11 = 2574
- 16. We put the2 and the 4 at the ends. We add the first pair 2 + 3 = 5. and we add the last pair: 3 + 4 = 7. Multiply by 11: 1) 151 = 2) 527 = 3) 333 = 4) 714 = 5) 909 = (5) Method for diving by 9. 23 / 9 = 2 remainder 5 The first figure of 23 is 2, and this is the answer. The remainder is just 2 and 3 added up! ● 43 / 9 = 4 remainder 7 The first figure 4 is the answer and 4 + 3 = 7 is the remainder - could it be easier? Divide by 9: 1) 61 = remainder 2) 33 = remainder 3) 44 = remainder 4) 53 = remainder ● 134 / 9 = 14 remainder 8 The answer consists of 1,4 and 8.1 is just the first figure of 134.4 is the total of the first two figures 1+ 3 = 4,and 8 is the total of all three figures 1+ 3 + 4 = 8. Divide by 9: 6) 232 = 7) 151 = 8) 303 = 9) 212 = remainder remainder remainder remainder ● 842 / 9 = 812 remainder 14 = 92 remainder 14 Actually a remainder of 9 or more is not usually permitted because we are trying to find how many 9's there are in 842. Since the remainder, 14 has one more 9 with 5 left over the final answer will be 93 remainder 5 Divide these by 9: 1) 771 2) 942 3) 565 4) 555 5) 777 6) 2382 (6) SOME BASIC RELATIONS AND REVIEW OF SQUARE AND CUBE FORMULA: Relations : If a = b , b = c then a= c If a > b , b > c then a> c If a < b , b < c then a < c If a b< b c then a < c If ab > bc then b > c If a > b , c> d then a + c> b+ d If a > b , c< d then a -c> b –d If a < b , c< d then a -c< b -d If a > b & a ,b both positive then 1/a < 1/b x = n then x = n or -n 2 2 (a + b) = a + 2ab + b 2 2 2
- 17. (a – b)= a –2ab + b 2 2 2 (a + b) = a + 3a b + 3ab + b 3 3 2 2 3 (a - b) = a -3a b + 3ab -b 3 3 2 2 3 (a + b+ c) = a + b + c + 2ab + 2bc+ 2ac 2 2 2 2 Factorization a –b = ( a + b )(a –b) 2 2 a + b = (a + b) ( a -ab + b ) 3 3 2 2 a -b = (a -b ) (a + ab+ b ) 3 3 2 2 Identities (a+ b) + (a –b) = 2(a + b ) 2 2 2 2 (a+ b) -(a–b) = 4ab 2 2 In dices a m xa n =a (m+n) a m /a n =a (m-n) a=1 0 a =1/a -m m a = m√a 1/m (a *b ) = amx b m m (a /b ) = a m m /b m (a ) = a m n m*n Logarithms a = n then log a n= x x log a (mn )= log am + log an log a (m/n) = log a m -log an log a (m) = n log am n log b n= log an/log ab Surd ( √a ) = a n n n√ a * √ b = √ab n n n√ a / √ b = √a /b n n m√ √ a = √a = √ √ a n mn n m m√ √ a = √a = √a p p mp p m Angle Measurement Total of Interior Angles in Degrees Triangle 180 Rectangle 360 Square 360 Pentagon 540 Circle 360 Some Facts In a triangle, interior opposite angle is always less than exterior angle.
- 18. Sum of 2interior opposite angles of a triangle is always equal to exterior angle. Triangle can have at one most obtuse angle. Angle made by altitude of a triangle with side on which it is drawn is equal to 90 degrees. in parallelogram opposite angles are equal. Squares Number Square Number Square Number Square 1 1 11 121 21 441 2 4 12 144 22 484 3 9 13 169 23 529 4 16 14 196 24 576 5 25 15 225 25 625 6 36 16 256 26 676 7 49 17 289 27 729 8 64 18 324 28 784 9 81 19 361 29 841 10 100 20 400 30 900 Cubes & Other Powers Number ( X) X3 X4 X5 1111 2 8 16 32 3 27 81 243 4 64 256 1024 5 125 625 3125 6 216 1296 7776 7 343 2401 16807 8 512 4096 32768 9 729 6561 59049 10 1000 10000 100000 (7) SQUARE ROOTS AND MULTIPLICATION FORMULA : No Roots No Roots No Roots No Roots 1 1 5 2.23 9 3 36 6 2 1.41 6 2.44 10 3.16 49 7 3 1.73 7 2.64 16 4 64 8 4 2 8 2.82 25 5 81 9 1 to 10 : 1 2 3 4 5 6 7 8 9 10 1 1 2 3 4 5 6 7 8 9 10 2 2 4 6 8 10 12 14 16 18 20 3 3 6 9 12 15 18 21 24 27 30 4 4 8 12 16 20 24 28 32 36 40 5 5 10 15 20 25 30 35 40 45 50 6 6 12 18 24 30 36 42 48 54 60 7 7 14 21 28 35 42 49 56 63 70 8 8 16 24 32 40 48 56 64 72 80 9 9 18 27 36 45 54 63 72 81 90 10 10 20 30 40 50 60 70 80 90 100 11to 20
- 19. 11 12 1314 15 16 17 18 19 20 1 11 12 13 14 15 16 17 18 19 20 2 22 24 26 28 30 32 34 36 38 40 3 33 36 39 42 45 48 51 54 57 60 4 44 48 52 56 60 64 68 72 76 80 5 55 60 65 70 75 80 85 90 95 100 6 66 72 78 84 90 96 102 108 114 120 7 77 84 91 98 105 112 119 126 133 140 8 88 96 104 112 120 128 136 144 152 160 9 99 108 117 126 135 144 153 162 171 180 10 110 120 130 140 150 160 170 180 190 200 (8) TWO INTERESTING FACTS: (9) ROMAN NUMERALS (10) NUMER REPRESENTATION: (11) DECIMAL REPRENTATION AND PREFIXS: (12) POWER OF INTIGERS: (13) SOME PROOFS BUT WITHOUT WORDS: (14) THE LARGEST PRIME NUMBERS DISCOVERED SO FAR: THE END Edited by Rakes Prasad ,2008 Contains: 1. Some different techniques of squaring ending with 1,2,3,4,6,7,8,9,etc . 2. adding different number sequence 3. finding the difference of squares. 4. dividing a 6-digit/3-digit number by 13,7,37037,11,41,15873 etc 5. dividing numbers by 75,125,625,12 2/3 etc.. 6. know why divisibility rules works? 7. finding percentage quickly. 8. various types of multiplication by 99,72,84,……… 9. squaring of some special numbers Rakes Prasad 2008. Squaring a 2-digit number ending in 1
- 20. .Take a 2-digitnumber ending in 1. Subtract 1 from the number. Square the difference. Add the difference twice to its square. Add 1. Example: If the number is 41, subtract 1: 41 - 1 = 40. 40 x 40 = 1600 (square the difference). 1600 + 40 + 40 = 1680 (add the difference twice to its square). 1680 + 1 = 1681 (add 1). So 41 x 41 = 1681. See the pattern? For 71 x 71, subtract 1: 71 - 1 = 70. 70 x 70 = 4900 (square the difference). 4900 + 70 + 70 = 5040 (add the difference twice to its square). . So 71 x 71 = 5041. Squaring a 2-digit number ending in 2 I Take a 2-digit number ending in 2. f The last digit will be _ _ _ 4. t Multiply the first digit by 4: the 2nd number will be h the next to the last digit: _ _ X 4. e n Square the first digit and add the number carried from u the previous step: X X _ _. m b e Example: r i s 52, the last digit is _ _ _ 4. 4 x 5 = 20 (four times the first digit): _ _ 0 4. 5 x 5 = 25 (square the first digit), 25 + 2 = 27 (add carry): 2 7 0 4. For 82 x 82, the last digit is _ _ _ 4. 4 x 8 = 32 (four times the first digit): _ _ 2 4. 8 x 8 = 64 (square the first digit), 64 + 3 = 67 (add carry): 6 7 2 4. Squaring a 2-digit number ending in 3 Take a 2-digit number ending in 3. 2 The last digit will be _ _ _ 9. 3 Multiply the first digit by 6:
- 21. the 2nd numberwill be the next to the last digit: _ _ X 9. 4 Square the first digit and add the number carried from the previous step: X X _ _. Example: 1 If the number is 43, the last digit is _ _ _ 9. 2 6 x 4 = 24 (six times the first digit): _ _ 4 9. 3 4 x 4 = 16 (square the first digit), 16 + 2 = 18 (add carry): 1 8 4 9. 4 So 43 x 43 = 1849. See the pattern? 1 For 83 x 83, the last digit is _ _ _ 9. 2 6 x 8 = 48 (six times the first digit): _ _ 8 9. 38 x 8 = 6 4 ( s q u a r e t he first digit), 64 + 4 = 68 (add carry): 6 8 8 9. 4 So 83 x 83 = 6889. Squaring a 2-digit number ending in 4 1 Take a 2-digit number ending in 4. 2 Square the 4; the last digit is 6: _ _ _ 6 (keep carry, 1.) 3 Multiply the first digit by 8 and add the carry (1); the 2nd number will be the next to the last digit: _ _ X 6 (keep carry). 4 Square the first digit and add the carry: X X _ _. Example: 1 If the number is 34, 4 x 4 = 16 (keep carry, 1); the last digit is _ _ _ 6. 2 8 x 3 = 24 (multiply the first digit by 8), 24 + 1 = 25 (add the carry): the next digit is 5: _ _ 5 6. (Keep carry, 2.) 3 Square the first digit and add the
- 22. carry, 2: 11 5 6. 4 So 34 x 34 = 1156. See the pattern? 1 For 84 x 84, 4 x 4 = 16 (keep carry, 1); the last digit is _ _ _ 6. 2 8 x 8 = 64 (multiply the first digit by 8), 64 + 1 = 65 (add the carry): the next digit is 5: _ _ 5 6. (Keep carry, 6.) Square the first digit and add the carry, 6: 7 0 5 6. So 84 x 84 = 7056. See previous chapters… 1 Choose a 2-digit number ending in 6. 2 Square the second digit (keep the carry): the last digit of the answer is always 6: _ _ _ 6 3 Multiply the first digit by 2 and add the carry (keep the carry): _ _ X _ 4 Multiply the first digit by the next consecutive number and add the carry: the product is the first two digits: XX _ _. Example: 1 If the number is 46, square the second digit : 6 x 6 = 36; the last digit of the answer is 6 (keep carry 3): _ _ _ 6 2 Multiply the first digit (4) by 2 and add the carry (keep the carry): 2 x 4 = 8, 8 + 3 = 11; the next digit of the answer is 1: _ _ 1 6 3 Multiply the first digit (4) by the next number (5) and add the carry: 4 x 5 = 20, 20 + 1 = 21 (the first two digits): 2 1 _ _ 4 So 46 x 46 = 2116. See the pattern? 1 For 76 x 76, square 6 and keep the carry (3): 6 x 6 = 36; the last digit of the answer is 6: _ _ _ 6 2 Multiply the first digit (7) by 2 and add the carry:
- 23. 2 x 7= 14, 14 + 3 = 17; the next digit of the answer is 7 (keep carry 1): _ _ 7 6 3 Multiply the first digit (7) by the next number (8) and add the carry: 7 x 8 = 56, 56 + 1 = 57 (the first two digits: 5 7 _ _ 4 So 76 x 76 = 5776. 1 Choose a 2-digit number ending in 7. 2 The last digit of the answer is always 9: _ _ _ 9 3 Multiply the first digit by 4 and add 4 (keep the carry): __X_ 4 Multiply the first digit by the next consecutive number and add the carry: the product is the first two digits: XX _ _. Example: 1 If the number is 47: 2 The last digit of the answer is 9: _ _ _ 9 3 Multiply the first digit (4) by 4 and add 4 (keep the carry): 4 x 4 = 16, 16 + 4 = 20; the next digit of the answer is 0 (keep carry 2): _ _ 0 9 4 Multiply the first digit (4) by the next number (5) and add the carry (2): 4 x 5 = 20, 20 + 2 = 22 (the first two digits): 2 2 _ _ 5 So 47 x 47 = 2209. See the pattern? 1 For 67 x 67 2 The last digit of the answer is 9: _ _ _ 9 3 Multiply the first digit (6) by 4 and add 4 (keep the carry): 4 x 6 = 24, 24 + 4 = 28; the next digit of the answer is 0 (keep carry 2): _ _ 8 9 4 Multiply the first digit (6) by the next number (7) and add 5 the carry (2): 6 x 7 = 42, 42 + 2 = 44 (the first two digits): 4 6 So 67 x 67 = 4489.
- 24. 1 Choose a2-digit number ending in 8. 2 The last digit of the answer is always 4: _ _ _ 4 3 Multiply the first digit by 6 and add 6 (keep the carry): __X_ 4 Multiply the first digit by the next consecutive number and add the carry: the product is the first two digits: XX _ _. Example: 1 If the number is 78: 2 The last digit of the answer is 4: _ _ _ 4 3 Multiply the first digit (7) by 6 and add 6 (keep the carry): 7 x 6 = 42, 42 + 6 = 48; the next digit of the answer is 8 (keep carry 4): _ _ 8 4 4 Multiply the first digit (7) by the next number (8) and add the carry (4): 7 x 8 = 56, 56 + 4 = 60 (the first two digits): 6 0 _ _ 5 So 78 x 78 = 6084. See the pattern? 1 For 38 x 38 The last digit of the answer is 4: _ _ _ 4 Multiply the first digit (3) by 6 and add 6 (keep the carry): 3 x 6 = 18, 18 + 6 = 24; the next digit of the answer is 4 (keep carry 2): _ _ 4 4 Multiply the first digit (3) by the next number (4) and add the carry (2): 3 x 4 = 12, 12 + 2 = 14 (the first two digits): 1 4 _ _ So 38 x 38 = 1444 Squaring a 2-digit number ending in 9 1 Choose a 2-digit number ending in 9. 2 The last digit of the answer is always 1. Multiply the first digit by 8 and add 8 (keep the carry): _ _ X _3
- 25. Multiply the firstdigit by the next consecutive number and add the carry: the product is the first two digits: XX _ _. Example: 1 If the number is 39: 2 The last digit of the answer is 1: _ _ _ 13 Multiply the first digit (3) by 8 and add 8 (keep the carry): 8 x 3= 24, 24 + 8 = 32; the next digit of the answer is 2 (keep carry 3): _ _ 2 1 4 Multiply the first digit (3) by the next number (4) and add the carry (3): 3 x 4 = 12, 12 + 3 = 15 (the first two digits): 1 5 _ _ 5 So 39 x 39 = 1521. rn? 1 For 79 x 79 2 The last digit of the answer is 1: _ _ _ 1 Multiply the first digit (7) by 8 and add 8 (keep the carry): 8 x 7 = 56, 56 + 8 = 64; the next digit of the answer is 4 (keep carry 6): _ _41 Multiply the first digit (7) by the next number (8) and add the carry (6) : 7 x 8 = 56, 56 + 6 = 62 (the first two digits): 6 2 _ _ So 79 x 79 = 6241. .. Choose two 2-digit numbers less than 20 (no limits for experts). Add all the numbers between them: 1 Add the numbers; 2 Subtract the numbers and add 1; 3 Multiply half the sum by this difference + 1, OR Multiply the sum by half the difference + 1. Example:
- 26. 1 If thetwo numbers selected are 6 and 19: 2 Add the numbers: 6 + 19 = 25. 3 Subtract the numbers: 19 - 6 = 13. Add 1: 13 + 1 = 14. 4 Multiply 25 by half of 14: 25 x 7 = 175. 5 So the sum of the numbers from 6 through 19 is 175. Therefore 6+7+8+9+10+11+12+13+14+15+16+17+18+19=175 See the pattern? 1 If the two numbers selected are 4 and 18: 2 Add the numbers: 4 + 18 = 22. 3 Subtract the numbers: 18 - 4 = 14. Add 1: 14 + 1 = 15. Multiply half of 22 by 15: 11 x 15 = 165 (10 x 15 + 15). So the sum of the numbers from 4 through 18 is 165. 1 Choose a 2-digit odd number. Add all the odd numbers starting with one through this 2-digit number: 2 Add one to the 2-digit number. 3 Divide this sum by 2 (take half of it). 4 Square this number. This is the sum of all odd numbers from 1 through the 2-digit number chosen. Example: 1 If the 2-digit odd number selected is 35: 2 35+1 = 36 (add 1). 3 36/2 = 18 (divide by 2) or 1/2 x 36 = 18 (multiply by 1/2). 4 18 x 18 = 324 (square 18): 18 x 18 = (20 - 2)(18) = (20 x 18) - (2 x 18) = 360 - 36 = 360 -30 - 6 = 324. 5 So the sum of all the odd numbers from one through 35 is 324. See the pattern? 1 If the 2-digit odd number selected is 79: 2 79+1 = 80 (add 1). 3 80/2 = 40
- 27. (divide by 2)or 1/2 x 80 = 40 (multiply by 1/2). 4 40 x 40 = 1600 (square 40). 5. So the sum of all the odd numbers from one through 79 is 1600. 1 Have a friend choose a a single digit number. (No restrictions for experts.) 2 Ask your friend to jot down a series of doubles (where the next term is always double the preceding one), and tell you the last term. 3 Ask your friend to add up all these terms. 4 You will give the answer before he or she can finish: The sum of all the terms of this series will be two times the last term minus the first term. Example: if the number selected is 9: 1 The series jotted down is: 9, 18, 36, 72, 144. 2 Two times the last term (144) minus the first (9): 2 x 144 = 288; 288 - 9 = 279. 3 So the sum of the doubles from 9 through 144 is 279. See the pattern? Here's one for the experts: 1 The number selected is 32: 2 The series jotted down is: 64, 128, 256, 512. 3 Two times the last term (512) minus the first (64): 2 x 512 = 1024; 1024 - 32 = 1024 - 30 - 2 = 994 - 2 = 992. 4 So the sum of the doubles from 32 through 512 is 992. Remember to subtract in steps from left to right. With practice you will be expert in summing series. restrictions for experts.) Ask your friend to jot down a series of
- 28. quadruples (where thenext term is always four times the preceding one), and tell you only the last term. Ask your friend to add up all these terms. You will give the answer before he or she can finish: The sum of all the terms of this series will be four times the last term minus the first term, divided by 3. Example: If the number selected is 5: 1 The series jotted down is: 5, 20, 80, 320, 1280. 2 Four times the last term (1280) minus the first (5): 4000 + 800 + 320 - 5 = 5120 - 5 = 5115 Divide by 3: 5115/3 = 1705 3 So the susum of the quadruples from 5 through 1280 is 1705. See the pattern? Here's one for the experts: 1 The number selected is 32: 2 The series jotted down is: 32, 128, 512, 2048. 3 Four times the last term (2048) minus the first (32): 8000 + 160 + 32 - 32 = 8,160 Divide by 3: 8160/3 = 2720. 4. So the sum of the quadruples from 32 through 2048 is 2720. Practice multiplying from left to right and dividing by 3. With practice you will be an expert quad adder. Add a sequence from one to a selected 2-digit number 1 Choose a 2-digit number. 2 Multiply the 2-digit number by half the next number, orMultiply half the 2-digit number by the next number.
- 29. Example: 1 If the2-digit even number selected is 51: 2 The next number is 52. Multiply 51 times half of 52. 3 51 x 26: (50 x 20) + (50 x 6) + 1 x 26) = 1000 + 300 + 26 = 1326 4 So the sum of all numbers from 1 through 51 is 1326. See the pattern? 1 If the 2-digit even number selected is 34: 2 The next number is 35. Multiply half of 34 x 35. 3 17 x 35: (10 x 35) + (7 x 30) + (7 x 5) = 350 + 210 + 35 = 560 + 35 = 595 4 So the sum of all numbers from 1 through 34 is 595. With some multiplication practice you will be able to find these sums of sequential numbers easily and faster than someone using a calculator! 1 Choose a 1-digit number. 2 Square it. Example: 1 If the 1-digit number selected is 7: 2 To add 1 + 2 + 3 + 4 + 5 +6+7+6+ 5 + 4 + 3 + 2 + 1 3 Square 7: 49 4 So the sum of all numbers from 1 through 7 and back is 49. See the pattern? 1 If the 1-digit number selected is 9: 2 To add 1 + 2 + 3 + 4 + 5 +6+7+8+ 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 3 Square 9: 81 4 So the sum of all numbers from 1 through 9 and back is 81. Add sequences of numbers in the 10's 1 Choose a 2-digit number in the 10's. To add all the 10's from 10 up
- 30. through this numberand down from it: 2 Square the 2nd digit of the number (keep the carry) _ _ X 3 The number of terms is 2 x the 2nd digit + 1. 4 The first digits = the number of terms (+ carry). nbsp; X X _ Example: 1 If the 2-digit number in the 10's selected is 16: (10 + 11 + 12 ... 16 + 15 + 14 ... 10) 2 Square the 2nd digit of the number: 6 x 6 = 36 (keep carry 3) _ _ 6 3 No. of terms = 2 x 2nd digit + 1: 2 x 6 + 1 = 13 4 No. of terms (+ carry): 13 + 3 = 16 1 6 _ 5 So the sum of the sequence is 166. See the pattern? 1. If the 2-digit number in the 10's selected is 18: (10 + 11 + 12 + ... 18 + 17 + 16 + 15 ... 10) 1 Square the 2nd digit of the number: 8 x 8 = 64 (keep carry 6) _ _ 4 2 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 17 3 No. of terms (+ carry): 17 + 6 = 23 2 3 _ 4 So the sum of the sequence is 234. 1 Choose a 2-digit number in the 20's. To add all the 20's from 20 up through this number and down from it: 2 Square the 2nd digit of the number (keep the carry) _ _ X 3 The number of terms is 2 x the 2nd digit + 1. 4 Multiply the number of terms by 2 (+ carry). X X _ Example: 1 If the 2-digit number in the 20's selected is 23: (20 + 21 + 22 + 23 + 22 + 21 + 20) 2 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 3 No. of
- 31. terms = 2x 2nd digit + 1: 2 x 3 + 1 = 7 4 2 x no. of terms: 2 x 7 = 14 1 4 _ 5 So the sum of the sequence is 149. See the pattern? 1 If the 2-digit number in the 20's selected is 28: (20 + 21 + 22 ... + 28 + 27 + ... 22 + 21 + 20) 2 Square the 2nd digit of the number: 8 x 8 = 64 (keep carry 6) _ _ 4 3 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 17 4 2 x no. of terms (+ carry): 2 x 17 + 6 = 40 4 0 _ 5 So the sum of the sequence is 404. 2 S 1 Choose a 2-digit number in the 30's. To add all the 30's qfrom 30 up through this number and down from it: u are the 2nd digit of the number (keep the carry) _ _ X 3 The number of terms is 2 x the 2nd digit + 1. 4 Multiply the number of terms by 3 (+ carry) X X _ Example: 1 If the 2-digit number in the 30's selected is 34: (30 + 31 + 32 + 33 + 34 + 33 + 32 + 31 + 30) 2 Square the 2nd digit of the number: 4 x 4 = 16 (keep carry 1) _ _ 6 3 No. of terms = 2 x 2nd digit + 1: 2 x 4 + 1 = 9 4 3 x no. of terms: 3 x 9 + 1 = 28 2 8 _ 5 So the sum of the sequence is 286. See the pattern? 1 If the 2-digit number in the 30's selected is 38: (30 + 31 + 2 32 + ... + 38 + 37 + ... 32 + 31 + 30) 3 Square the 2nd digit of the number: 8 x 8 = 64 (keep carry 6) _ _ 4
- 32. No. of terms= 2 x 2nd digit + 1: 2 x 8 + 1 = 17 3 x no. of terms: 3 x 17 + 6 = 51 + 6 = 57 5 7 _ So the sum of the sequence is 574. 1 Choose a 2-digit number in the 40's. To add all the 40's from 40 up through this number and down from it: 2 Square the 2nd digit of the number (keep the carry) _ _ X 3 The number of terms is 2 x the 2nd digit + 1. 4 Multiply the number of terms by 4 (+ carry) X X _ e 2-digit number in the 40's selected is 43: (40 + 41 + 42 + 43 + 42 + 41 + 40) 1 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 2 No. of terms = 2 x 2nd digit + 1: 2 x 3 + 1 = 7 3 4 x no. of terms: 4 x 7 = 28 2 8 _ 4 So the sum of the sequence is 289. See the pattern? 1 If the 2-digit number in the 40's selected is 48: (40 + 41 + 42 + ... + 48 + 47 + ... 42 + 41 + 40) 2 Square the 2nd digit of the number: 8 x 8 = 64 (keep carry 6) _ _ 4 3 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 17 4 4 x no. of terms: 4 x 17 + 6 = 40 + 28 + 6 = 68 + 6 = 74 7 4 _ 5 So the sum of the sequence is 744. 1 Choose a 2-digit number in the 50's. To add all the 50's from 50 up through this number and down from it: 2 Square the 2nd digit of the number (keep the carry) _ _ X 3 The number of terms is 2 x the 2nd digit + 1. 4 Multiply the number of terms by 5. X X _ Example:
- 33. 1 If the2-digit number in the 50's selected is 53: (50 + 51 + 52 + 53 + 52 + 51 + 50) 2 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 3 No. of terms = 2 x 2nd digit + 1: 2 x 3 + 1 = 7 4 5 x no. of terms: 5 x 7 = 35 3 5 _ 5 So the sum of the sequence is 359. See the pattern? 1 If the 2-digit number in the 50's selected is 57: (50 + 51 + 5 2 ... + 57 + 56 + ... 52 + 51 + 50) 2 Square the 2nd digit of the number: 7 x 7 = 49 (keep carry 4) _ _ 9 3 No. of terms = 2 x 2nd digit + 1: 2 x 7 + 1 = 15 5 x no. of terms (+ carry): 5 x 15 + 4 = 75 + 4 = 79 7 9 _ So the sum of the sequence is 799. Add sequences of numbers in the 60's 1 Choose a 2-digit number in the 60's. To add all the 60's from 60 up through this number and down from it: 2 Square the 2nd digit of the number (keep the carry) _ _ X 3 The number of terms is 2 x the 2nd digit + 1. 4 Multiply the number of terms by 6 (+ carry) X X _ Example: 1 If the 2-digit number in the 60's selected is 63: (60 + 61 + 62 + 63 + 62 + 61 + 60) 2 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 3 No. of terms = 2 x 2nd digit + 1: 2 x 3 + 1 = 7 4 5 6 x no. of terms: 6 x 7 = 42 4 2 _ So the sum of the sequence is 429. See the pattern? 1 If the 2-digit number in the 60's selected is 68: (60 + 61 + 62 + ... + 68 + 67 + ... 62 + 61 + 60) 2 Square the 2nd digit of
- 34. the number: 8x 8 = 64 (keep carry 6) _ _ 4 3 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 17 4 6 x no. of terms: 6 x 17 + 6 = 42 + 1 = 102 + 6 = 108 1 0 8 _ 5oS the sum of the sequence is 1084. 1 Choose a 2-digit number in the 70's. To add all the 70's from 70 up through this number and down from it: 2 Square the 2nd digit of the number (keep the carry) _ _ X 3 The number of terms is 2 x the 2nd digit + 1. 4 Multiply the number of terms by 7 (+ carry) X X _ Example: 1 If the 2-digit number in the 70's selected is 73: (70 + 71 + 72 + 73 + 72 + 71 + 70) 2 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 3 No. of terms = 2 x 2nd digit + 1: 2 x 3 + 1 = 7 4 7 x no. of terms: 7 x 7 = 49 4 9 _ 5 So the sum of the sequence is 499. See the pattern? 1 If the 2-digit number in the 70's selected is 78: (70 + 71 + 72 + ... + 78 + 77 + ... 72 + 71 + 70) 2 Square the 2nd digit of the number: 8 x 8 = 64 (keep carry 6) _ _ 4 3 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 17 4 7 x no. of terms: 7 x 17 + 6 = 49 + 6 = 119 + 6 = 125 1 2 5 _ 5 So the sum of the sequence is 1254. 1 Choose a 2-digit number in the 80's. To add all the80's from 80 up through this number and down from it: 2 Square the 2nd digit of the number (keep the carry) _ _
- 35. umber of termsis 2 x the 2nd digit + 1 4 Multiply the number of terms by 8 (add the carry) X X _ Example: 1 If the 2-digit number in the 80's selected is 83: (80 + 81 + 82 + 83 + 82 + 81 + 80) 2 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 3 No. of terms = 2 x 2nd digit + 1: 2 x 3 + 1 = 7 4 8 x no. of terms: 8 x 7 = 56 5 6 _ 5 So the sum of the sequence is 569. See the pattern? 1 If the 2-digit number in the 80's selected is 88: (80 + 81 + 82 ... + 88 + 87 + ... 83 + 82 + 81 + 80) 2 Square the 2nd digit of the number: 8 x 8 = 64 (keep carry 6) _ _ 4 3 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 17 8 x no. of terms (+ carry): 8 x 17 + 6 = 80 + 56 + 6 = 136 + 6 = 42 4 2 _ So the sum of the sequence is 1424. 1 Have a friend choose and write down a single-digit number. (Two digits for experts.) 2 Ask your friend to name and note a third number by adding the first two. 3 Name a fourth by adding the second and third. Continue in this way, announcing each number, through ten numbers. 4 Ask your friend to add up the ten numbers. You will give the answer before he or she can finish: The sum of all the terms of this series will be the seventh number multiplied by 11.
- 36. Example: 1 If thenumbers selected are 7 and 4: e series jotted down is: 4, 7, 11, 18, 29, 47, 76, 123, 199, 322. 3 The seventh number is 76. 11 x 76 = 836 (use the shortcut for 11: 7 is the first digit, 6 is the third digit; the middle digit will be 7 + 6, and carry the 1: 836). 4 So the sum of the ten numbers is 836. Here are some of the calculations that you can do mentally after practicing the exercises. See how many you can do. Check your answers with a calculator. Your mental math powers should be impressive! 15 x 15 89 x 89 394 x 101 32 x 38 51 x 59 101 x 101 x 94 93 x 93 228 x 101 25 x 25 147 56 x 56 94 x 96 101 x 448 83 x 87 687 x 101 64 x 66 52 x 52 101 x 654 35 x 35 11 x 19 101 x 61 x 69 89 x 101 61 x 61 34 x 36 206 101 x 48 52 x 52 45 x 45 456 x 101 54 x 54 41 x 41 101 x 41 32 x 32 83 x 101 29 x 29 55 x 55 63 x 67 479 x 101 82 x 82 882 x 101 14 x 16 319 x 101 73 x 77 65 x 65 101 x 859 13 x 17 101 x 149 41 x 49 82 x 101 75 x 75 101 x 71 x 79 69 x 101 22 x 22 993 x 101 144 51 x 51 85 x 85 101 x 101 129 x 101 69 x 69 31 x 39 738 x 101 21 x 29 71 x 71 77 x 101 95 x 95 265 x 101 74 x 76 53 x 53 101 x 409 51 x 51 98 x 101 94 x 94 42 x 42 53 x 57 339 x
- 37. 62 x 68101 x 88 19 x 19 238 x 101 101 43 x 47 96 x 96 21 x 21 101 x 279 81 x 89 648 x 62 x 62 101 x 57 24 x 26 101 x 668 101 99 x 99 23 x 27 22 x 28 515 x 101 97 x 97 79 x 79 101 x 826 245 x 101 31 x 31 37 x 33 101 x 82 x 88 39 x 39 126 x 101 91 x 99 218 68 x 101 51 x 101 598 x 101 98 x 98 72 x 72 91 x 91 54 x 56 101 x 29 57 x 57 771 x 101 81 x 81 559 x 101 52 x 58 59 x 59 84 x 86 46 x 44 101 x 25 12 x 18 101 x 697 92 x 92 349 x 93 x 97 101 x 188 42 x 48 49 x 49 101 58 x 58 92 x 92 101 x 78 37 x 101 1 Select two consecutive 2-digit numbers. 2 Add the two 2- digit numbers! Examples: 1 24 + 25 = 49. (Try it on a calculator and see, or if you're really sharp, do it mentally: 24 x 24 = 576, 25 x 25 = 625, 625 - 576 = 49.) 2 If 63 and 64 are selected, then 63 + 64 = 127. (For larger number addition, do it in steps: 63 + 64 = 63 + 60 + 4 = 123 + 4 = 127.) 1 Select two consecutive 2-digit numbers, one not more than 10 larger than the other (experts need not use this limitation). 2 Subtract the smaller number from the larger. 3 Add the two numbers. 4 Multiply the first answer by the second. Examples:
- 38. 1 If 71and 64 are selected: 2 71 - 64 = 7. 3 71 + 64 = Add left to right: 71 + 64 = 71 + 60 + 4 = 131 + 4 = 135) 4 Multiply these results: 7 x 135 = 945 (Multiply left to right: 7 x 135 = 7 x (100+30+5) = 700 + 210 + 35 = 910 + 35 = 945) 5 So the difference of the squares of 71 and 64 is 945. See the pattern? 1 If 27 and 36 are selected: 3 2 36 - 27 = 9. 36 + 27 = 63 (Think: 27 + 30 + 6 = 57 + 6 = 63) Multiply these results: 9 x 63 = 567 (Think: 9 x (60+3) = 540 + 27 = 567) So the difference of the squares of 27 and 36 is 567. Why do the divisibility 'rules' work? From: cms@dragon.com (Cindy Smith) To: Dr. Math <dr.math@forum. swarthmore.edu> Subject: Factoring Tricks In reading a popular math book, I came across several arithmetic factoring tricks. Essentially, if the last digit of a number is zero, then the entire number is divisible by 10. If the last number is even, then the entire number is divisible by 2. If the last two digits are divisible by 4, then the whole number is. If the last three digits divide by 8, then the whole number does. If the last four digits divide by 16, then the whole number does, etc. If the last digit is 5, then the whole number divides by 5. Now for the tricky ones. If you add the digits in a number and the sum is divisible by 3, then the whole number is. Similarly, fif you add the digits in a number and the sum is divisible by 9, then the whole number is. For example, take the number 1233: 1 + 2 + 3 + 3 = 9.
- 39. Therefore, the wholenumber is divisible by 9 and the quotient is 137. The number is also divisible by 3 and the quotient is 411. It works for extremely large numbers too (I checked on my calculator). Now here's a really tricky trick. You add up the alternate digits of a number and then add up the other set of alternate digits. If the sums of the alternate digits equal each other then the whole number is divisible by 11. Also, if the difference of the alternate digits is 11 or a multiple of 11, then the whole number is divisible by 11. For example, 123,456,322. 1 + 3 + 5 + 3 + 2 = 14 and 2 + 4 + 6 + 2 = 14. Therefore, the whole number is divisible by 11 and the quotient is 11,223,302. Also, if a number is divisible by both 3 and 2, then the whole number is divisible by 6. The only single digit number for which there is no trick listed is 7. I find these rules interesting and useful, especially when factoring large numbers in algebraic expressions. However, I'm not sure why all these rules work. Can you explain to me why these math tricks work? If I could understand why they work, I think it would improve my math skills. Thanks in advance for your help. Cindy Smith cms@dragon.com From: Dr. Math <dr.math@forum.swarthmore.edu> To: cms@dragon. com (Cindy Smith) Subject: Re: Factoring Tricks Thank you for this long and very well-written question. I will try to write as clearly as you as I answer. All these digital tests for
- 40. divisibility are basedon the fact that our system of numerals is written using the base of 10. The digits in a string of digits making up a numeral are actually the coefficients of a polynomial with 10 substituted for the variable. For example, 1233 = 1*10^3 + 2*10^2 + 3*10^1 + 3*10^0 which is gotten from the polynomial 1*x^3 + 2*x^2 + 3*x^1 + 3*x^0 = x^3 + 2*x^2 + 3*x + 3 by substituting 10 for x. We can explain each of these tricks in terms of that fact: 10 - numerals ending in 0 represent numbers divisible by 10: Since the last digit is zero, and all other terms in the polynomial form are divisible by 10, the number is divisible by 10. Similarly, if the number is divisible by 10, since all the terms except the last one are automatically divisible by 10 no matter what the coefficients or digits are, the number will be divisible by 10 only if the last digit is. Since all the digits are smaller than 10, the last digit has to be 0 to be a multiple of 10.2 - numerals ending in an even digit represent numbers divisible by 2: f o Same argument as above about all the terms except the last one being divisible by 2. The last digit is
- 41. divisible by 2(even) if and only if the whole number is. R 4 - numerals ending with a two-digit multiple of 4 represent numbers divisible by 4:o f Similar to the above, but since 10 is not a multiple of 4, but 10^2 is, we have to look at the last two digits instead of just the last digit. a n 8 - numerals ending with a three-digit multiple of 8 represent numbers divisible by 8: u Similar to 4, but now 10^2 is not a multiple of 8, but 10^3 is, so we have to look at the last three mdigits. er a 16 - You figure this one! l 5 - You figure this one, too! ook 3 - numerals whose sum of digits is divisible by 3 represent numbers s divisible by 3: This one is different, because 3 does not divide any power of 10 evenly. That means we will have to consider the effect of all the l digits. Here we use this fact: 10^k - 1 = (10 - 1)*(10^(k- 1) = ... + 10^2 + i 10 + 1) This is a fancy way of saying 9999...999 = 9*1111...111 . We use k this to rewrite our powers of 10 as 10^k = 9*a[k] + 1. Now the polynomial e this: d[k]*10^k + d[k-1]*10^(k-1) + ... + d[1]*10 + d[0] = d[k ]*(9*a[k] + 1) + ... + d[1]*(9*a[1] + 1) + d[0] = 9*(d[k]*a[k] + ... + d[1]*a[1])
- 42. + d[k] +... + d[1] + d[0] Now notice that 3 divides the first part, so the whole number is divisible by 3 if and only if the sum of the digits is. 9 - numerals whose sum of digits is divisible by 9 represent numbers divisible by 9: Use the same equation as the previous case. You figure the rest! 11 - numerals whose alternating sum of digits is divisible by 11 represent numbers divisible by 11: Here the phrase "alternating sum" means we alternate the signs from positive to negative to positive to negative, and so on. We use this fact: 10 to an odd power plus 1 is divisible by 11, and 10 to an even power minus 1 is divisible by 11. The first part is a fancy way of writing 10000. ..0001 = 11*9090...9091 (where there are an even number of 0's on the left-hand side). The second part is a fancy way of writing 99999...9999 = 11*9090...0909 (where there are an even number of 9's in the left hand side). We write 10^(2*k) = 11*b[2*k] + 1 and 10^(2*k+1) = 11*b [2*k+1] - 1. Here k is any nonnegative integer. We substitute that into the polynomial form, so:d[2*k]*10^(2*k) + d[2*k- 1]*10^(2*k-1) + . .. + d[1]*10 + d[0] = d[2*k]*(11*b[2*k] + 1) + d[2*k-1]*(11*b[2*k-1] - 1) + ...
- 43. � + d[1]*(11*a[1]- 1) + d[0] = 11*(d[2*k]*b[2*k] + ... + d[1]*a[1]) + d[2*k] - d[2*k-1] � + ... - d[1] + d[0] The first part is divisible by 11 no matter what the digits are, so the whole number is divisible by 11 if and only if the last part, which is the alternating sum of the digits, is divisible by 11. If you prefer, you can write d[2*k] - d[2*k-1] + ... - d[1] + d[0] = (d[0] + d[2] + ... + d[2*k]) - (d[1] + d[3] + ... + d[2*k-1]), so that you add up every other digit, starting from the units digit, and then add up the remaining digits, and subtract the two sums. This will compute the same result as the alternating sum of the digits. 7 - There is a trick for 7 which is not as well known as the others. It makes use of the fact that 10^(6*k) - 1 is divisible by 7, and 10^(6*k - 3) + 1 is divisible by 7. It goes like this: Mark off the digits in groups of threes, just as you do when you put commas in large numbers. Starting from the right, compute the alternating sum of the groups as three-digit numbers. If the result is negative, ignore the sign. If the result is greater than 1000, do the same thing to the resulting number until you have a result between 0 and 1000 inclusive. That 3- digit
- 44. number is divisibleby 7 if and only if the original number is too. Example: 123471023473 = 123,471,023,473, so make the sum 473 - 23 + 471 - 123 = 450 + 348 = 798. 798 = 7*114, so 798 is divisible by 7, and 123471023472 is, too. An extra trick is to replace every digit of 7 by a 0, every 8 by a 1, and every 9 by a 2, before, during, or after the sum, and the fact remains. The sum could also have been computed as 473 - 23 + 471 - 123 --> 403 - 23 + 401 - 123 = 380 + 278 --> 310 + 201 = 511 = 7*73. You can figure out why this "casting out 7's" part works. There is another way of testing for 7 which uses the fact that 7 divides 2*10 + 1 = 21. Start with the numeral for the number you want to test. Chop off the last digit, double it, and subtract that from the rest of the number. Continue this until you get stuck. The result is 7, 0, or -7, if and only if the original number is a multiple of 7. Example: 123471023473 --> 12347102347 - 2*3 = 12347102341 --> 1234710234 - 2*1 = 1234710232 --> 123471023 - 2*2 = 123471019 --> 12347101 - 2*9 = 12347083 --> 1234708 - 2*3 = 1234702 --> 123470 - 2*2 = 123466 --> 12346 - 2*6 = 12334 --> 1233 - 2*4 = 1225 --> 122 - 2*5 = 112 --> 11 - 2*2 = 7.
- 45. 13 - Thesame trick that works for 7 works for 13; that is, 13 divides 10^(6*k) - 1 and 10^(6*k - 3) + 1, so the alternating sum of three-digit groups works here, too. 17 - This is harder. You would have to use alternating sums of 8-digit groups! 1 Select a 3-digit number. 2 Repeat these digits to make a 6- digit number. 3 Divide these 6 digits by 7, then by 13. 4 The answer i s 11 times the first three digits! Example: 1 If the 3-digit number selected is 234: 2 The 6-digit number is 234234. 3. Divide by 7, then by 13: multiply by 11 -to multiply 234 by 11, work right to left: last digit on right = _ __4 next digit to left = 3 + 4 = 7: _ _ 7 _ next digit to left = 2 + 3 = 5: _ 5 _ _ last digit on left = 2 _ _ _ 3 So 234234 divided by 7, then 13 is 2574. See the pattern? 1 If the 3-digit number selected is 461: 2 The 6-digit number is 461461. 3. Divide by 7, then by 13: multiply by 11 -to multiply 461 by 11, work right to left: last digit on right = _ __1 nextdigit to left = 4 + 6 = 10: _ 0 _ _ last digit on left = 4 + 1 (carry) = 5: 5 _ _ _ 3 So 461461 divided by 7, then 13 is 5071. Practice multiplying by 11 - this process works for multiplying any number by 11.
- 46. 1 Select a3-digit number. 2 Repeat these digits to make a 6- digit number. 3 Divide these 6 digits by 13, then by 11. 4 The answer is 7 times the first three digits! Example: 1 If the 3-digit number selected is 231: 2 The 6-digit number is 231231. 3 Divide by 13, then by 11: 7 x 231 = 1400 + 210 + 7 = 1617. 4 So 231231 divided by 13, then 11 is 1617. See the pattern? 1 If the 3-digit number selected is 412: 2 The 6-digit number is 412412. Reme mber to multiply left to right and add in increments. Then you will be able to give these answers quickly and accurately. 1 Select a 6-digit number repeating number. 2 Repeat these digits to make a 6-digit number. 3 Multiply a single digit by 3, then by 5. Example: 1 If the 6-digit repeating number selected is 333333: 2 Multiply 3 x 3: 9 3 Multiply 9 x 5: 45 4 So 333333 divided by 37037 and multiplied by 5 is 45. See the pattern? You can expand this exercise by using a different number in the final step. Example: multiply by 4: 1 If the 6-digit repeating number selected is 555555: 2 Multiply 3 x 5: 15. 3
- 47. Multiply 15 x4: 60 4 So 555555 divided by 37037 and multiplied by 4 is 60. By changing the last step you can generate many extensions of this 3 dsivide by 13, then by 11: 7 x 412 = 2800 + 70 + 14 = 2884. 4 So 412412 divided by 13, then 11 is 2884. exercise. Be inventive and create some impressive calculations. Dividing a repeating 6-digit number by 7, 11, 13; subtract 101 Select a 3-digit number. Repeat these digits to make a 6-digit number. Divide this 6-digit by 7, then 11, then 13. Subtract 101. The answer is the original number minus 101! Example: If the 3-digit number selected is 289: The 6-digit number is 289289. Divide by 7, then by 11, then by 13: the answer is 289. Subtract 101: 289 - 101 = 188 So 289289 divided by 7, then 11, then 13 minus 101 is 188. See the pattern? If the 3-digit number selected is 983: The 6-digit number is 983983. Divide by 7, then by 11, then by 13: the answer is 983. 1 Subtract 101: 983 - 101 = 882 So 983983 divided by 7, then 11, then 13 minus 101 is 882. 1 Select a repeating 3-digit number . 2 The answer is 3 times one of the digits plus 41! Example: 1 Select 999. 2 Multiply one digit by 3: 9 x 3 = 27. 3 Add 41: 27 + 41
- 48. = 68. 4So (999 / 37) + 41 = 68. Change the last step to add other numbers, and thus produce many new exercises. 1 Select a repeating 6-digit number . 2 The answer is 7 times the first digit of the number! Example: 1 777777 / 15873 = 7 x 7 = 49. 2 555555 / 15873 = 7 x 5 = 35. 3 999999 / 15873 = 7 x 9 = 63. Not very demanding mental math, but good for a quick challenge or two. Dividing mixed numbers by 2 1 Select a mixed number (a whole number and a fraction). � 2. If the whole number is even, divide by 2 - this is the whole number of the answer. �� The numerator of the fraction stays the same; multiply the denominator by 2. � 3. If the whole number is odd, subtract 1 and divide by 2 - this is the whole number of the answer. � Add the numerator and the denominator of the fraction - this will be the new numerator of the fraction; � Multiply the denominator by 2. Even whole number example: If the first number selected is 8 3/4: Divide the whole number (8) by
- 49. 2: 8/2 =4 (whole number) Use the same numerator: 3 Multiply the denominator by 2: 4 x 2 = 8 (denominator) So 8 3/4 divided by 2 = 4 3/8. Odd whole number example: 1 If the first number selected is 13 2/5: 2 3 Subtract 1 from the whole number and divide by 2: 13 - 1 = 12, 12/2 = 6. Add the numerator and the denominator: 2 + 5 = 7. This is the numerator of the fraction. Multiply the denominator by 2: 5 x 2 = 10 . So 13 2/5 divided by 2 = 6 7/10. file:/// 1 Select a 2-digit number. 2 Multiply it by 8 (or by 2 three times). 3 Move the decimal point 2 places to the left. Example: 1 The 2-digit number chosen to multiply by 12 1/2 is 78. 2 Multiply by 2 three times: 2 x 78 = 156 2 x 156 = 312 2 x 312 = 624 3 Move the decimal point 2 places to the left: 6.24 4 So 78 divided by 12 1/2 = 6.24. See the pattern? 1 If the 2-digit number chosen to multiply by 12 1/2 is 91: 2 Double three times: 182, 364, 728. 3 Move the decimal point 2 places to the left: 7.28 4 So 91 divided by 12 1/2 = 7.28.5 ividing a 2-digit number by 15 1 Select a 2-digit number. 2 Multiply it by 2. 3 Divide the result by 3. 4 Move the decimal point 1 place to the left. Example: 1 The 2-digit number chosen to multiply by 15 is 68. 2 Multiply by 2:
- 50. 2 x 68= 120 + 16 = 136 3 Divide the result by 3: 136/3 = 45 1/3 4 Move the decimal point 1 place to the left: 4.5 1/3 5 So 68 divided by 15 = 4.5 1/3. See the pattern? 1 The 2-digit number chosen to multiply by 15 is 96. 2 Multiply by 2: 2 x 96 = 180 + 12 = 192 3 Divide the result by 3: 192/3 = 64 4 Move the decimal point 1 place to the left: 6.4 5 So 96/15 = 6.4. With this method you will be able to divide numbers by 15 with two quick 1 Select a 2-digit number. (Choose larger numbers when you feel sure about the method.) 2 Multiply by 4 (or by 2 twice). 3 Move the decimal l point two places to the left. Example: 1 The 2-digit number chosen to divide by 25 is 38. 2 Multiply by 4: 4 x 38 = 4 x 30 = 120 + 32 = 152. 3 Move the decimal point 2 places to the left: 1.52 4 So 38 divided by 25 = 1.52. See the pattern? 1 The 3-digit number chosen to divide by 25 is 641. 2 Multiply by 2 twice: 2 x 64 = 1282. 2 x 1282 = 2400 + 164 = 2564. 3 Move the decimal point 2 places to the left: 25.64. 4 So 641 divided by 25 = 25.64. 1 Select a 2-digit number (progress to larger ones). 2 Multiply it by 3. 3 Move the decimal point 2 places to the left.
- 51. Example: 1 The 2-digitnumber chosen to multiply by 33 1/3 is 46. 2 Multiply by 3: 3 x 46 = 3(40 + 6) = 120 + 18 = 138 3 Move the decimal point 2 places to the left: 1.38 4 So 46 divided by 33 1/3 = 1.38. (If you divide by 33.3 using a calculator, you will not get the exact answer.) See the pattern? 1 If the 3-digit number chosen to multiply by 33 1/3 is 650: 2 Multiply by 3: 3 x (600 + 50) = 1800 + 150 = 1950 3 Move the decimal point 2 places to the left: 19.50 4 So 650 divided by 33 1/3 = 19.5. Practice multiplying left to right and this procedure will become an easy one - and you will get exact answers, too. Dividing a 2- or 3-digit number by 35 1 Select a 2-digit number. (Choose larger numbers when you feel sure about the method.) 2 Multiply by 2. 3 Divide the resulting number by 7. 4 Move the decimal point 1 place to the left. Example: If the number chosen to divide by 35 is 61: Multiply by 2: 2 x 61 = 122. Divide by 7: 122/7 = 17 3/7 Move the decimal point 1 place to the left: 1.7 3/7 So 61 divided by 35 = 1.7 3/7. See the pattern? 1 If the number chosen to divide by 35 is 44: 2 Multiply by 2: 2 x 44 = 88 3 Divide by 7: 88/7 = 12 4/7 4 Move the decimal point 1 place to the left: 1.2 4/7 5 So 44 divided by 35 = 1.2 4/7.
- 52. Division done bycalculator will give repeating decimals (unless the original number is a multiple of 7), truncated by the limits of the display. The exact answer must be expressed as a mixed number. 1 Select a 2-digit number. 2 Multiply by 8. 3 Divide the product by 3. 4 Example: 1 The 2-digit number chosen to divide by 37 1/2 is 32. 2 Multiply by 8: 8 x 32 = 240 + 16 = 256 3 Divide by 3: 256/3 = 85 1/3 4 Move the decimal point two places to the left: .85 1/3 5 So 32 divided by 37 1/2 = .85 1/3. See the pattern? 1 The 2-digit number chosen to divide by 37 1/2 is 51. 2 Multiply by 8: 8 x 51 = 408 3 Divide by 3: 408/3 = 136 4 Move the decimal point two places to the left: 1.36 5 So 51 divided by 37 1/2 = 1.36. Dividing a 2-digit number by 45 1 Select a 2-digit number. 2 Divide by 5. 3 Divide the resulting number by 9.Example: 1 f the number chosen to divide by 45 is 32: 2 Divide by 5: 32/5 = 6.4 3 Divide the result by 9: 6.4/9 = .71 1/9 4 So 32 divided by 45 = .71 1/9. See the pattern? 1 If the number chosen to divide by 45 is 61: 2 Divide by 5: 61/5 = 12.2 3 Divide the result by 9: 12.2/9 = 1.35 5/9 4 So 61 divided by 45 1.35 5/9.
- 53. 1 Select a2-digit number. (Choose larger numbers when you feel sure about the method.) 2 Multiply by 4 (or by 2 twice). 3 Move the decimal point two places to the left. 4 Divide by 3 (express remainder as a fraction). Example: 1 The 2-digit number chosen to divide by 75 is 82. 2 Multiply by 4: 4 x 82 = 328. 3 Move the decimal point 2 places to the left: 3.28 4 Divide by 3: 3.28/3 = 1.09 1/3 5 So 82 divided by 75 = 1.09 1/3. See the pattern? 1 The 3-digit number chosen to divide by 75 is 631. 2 Multiply by 4 (multiply left to right): 4 x 631 = 2400 + 120 + 4 = 2520 + 4 = 2524. 3 Move the decimal point 2 places to the left: 25.24. 4 Divide by 3: 25.24/3 = 8.41 1/3 5 So 631 divided by 75 = 8.41 1/3. Select a number. 2 Multiply it by 2 3 Divide the result by 3. Example: 1 The number chosen to divide by 1 1/2 is 72. 2 Multiply by 2: 2 x 72 = 144 3 Divide by 3: 144 / 3 = 48 4 So 72 divided by 1 1/2 = 48. See the pattern? 1 The number chosen to divide by 1 1/2 is 83. 2 Multiply by 2: 2 x 83 = 166 3 Divide by 3: 166 / 3 = 55 1/3 4 So 83 divided by 1 1/2 = 55 1/3. 1 Select a 2-digit number. 2 Multiply by 3. 3 Divide by 4. Example:
- 54. 1 The 2-digitnumber chosen to divide by 1 1/3 is 47. 2 Multiply by 3: 3 x 47 = 120 + 21 = 141 3 Divide by 4: 141/4 = 35 1/4 4 So 47 divided by 1 1/3 = 35 1/4. See the pattern? 1 The 2-digit number chosen to divide by 1 1/3 is 82. 2 Multiply by 3: 3 x 82 = 246 3 Divide by 4: 246/4 = 61 1/2 4 So 82 divided by 1 1/3 = 61 1/2. With this pattern you will be able to give answers quickly, but most importantly, your answers will be exact. If a calculator user divides by 1.3, the answer will NOT be correct. 1 Select a 2-digit number. 2 Multiply by 8. 3 Move the decimal point one place to the left. Example: 1 The 2-digit number chosen to divide by 1 1/4 is 32. 2 Multiply by 8: 8 x 32 = 240 + 16 = 256 3 Move the decimal point one place to the left : 25.6 4 So 32 divided by 1 1/4 = 25.6. See the pattern? 1 The 2-digit number chosen to divide by 1 1/4 is 64. 2 Multiply by 8: 8 x 64 = 480 + 32 = 512 3 Move the decimal point one place to the left: 51.2 4 So 64 divided by 1 1/4 = 51.2. Multiply from left to right for ease and accuracy. You will soon be doing this division by a mixed number quickly. 1 Select a number. 2 Multiply it by 5 3 Divide the result by 6. Example:
- 55. 1 The numberchosen to divide by 1 1/5 is 24. 2 Multiply by 5: 5 x 24 = 120 3 Divide by 6: 120/6 = 20 4 So 24 divided by 1 1/5 = 20. See the pattern? 1 The number chosen to divide by 1 1/5 is 76. 2 Multiply by 5: 5 x 76 = 350 + 30 = 380 3 Divide by 6: 380/6 = 63 2/6 4 So 76 divided by 1 1/5 = 63 1/3. Dividing a 2-digit number by 1 2/3 1 Select a 2-digit number. 2 Multiply by 6 (or by 2 and 3). 3 Move the decimal point one place to the left. Example: 1 The 2-digit number chosen to divide by 1 2/3 is 78. 2 Multiply by 3: 3 x 78 = 210 + 24 = 234 3 Multiply by 2: 2 x 234 = 468 4 Move the decimal point one place to the left: 46.8 5 So 78 divided by 1 2/3 = 46.8. See the pattern? 1 The 2-digit number chosen to divide by 1 2/3 is 32. 2 Multiply by 3: 3 x 32 = 96 3 Multiply by 2: 2 x 96 = 180 + 12 = 192 4 Move the decimal point one place to the left: 19.2 5 So 32 divided by 1 2/3 = 19.2. Practice multiplying from left to right and you will become adept at mentally dividing a number by 1 2/3. 1 Select a 2-digit number. 2 Multiply by 8. 3 Move the decimal point 3 places to the left. Example:
- 56. 1 The 2-digitnumber chosen to divide by 125 is 72. 2 Multiply by 8: 8 x 72 = 560 + 16 = 576. 3 Move the decimal point 3 places to the left : .576 4 So 72 divided by 125 = .576. See the pattern? 1 The 2-digit number chosen to divide by 125 is 42. 2 Multiply by 8: 8 x 42 = 320 + 16 = 336. 3 ove the decimal point 3 places to the left: .336 4 So 42 divided by 125 = .336. 1 Select a 2-digit number. 2 Multiply by 4. 3 Divide the product by 7. Example: 1 The 2-digit number chosen to divide by 1 3/4 is 34. 2 Multiply by 4: 4 x 34 = 136 3 Divide the product by 7: 137/6 = 19 3/7 (If you use a calculator, you will get a long, inexact decimal number.) 4 So 34 divided by 1 3/4 = 19 3/7. See the pattern? 1 The 2-digit number chosen to divide by 1 3/4 is 56. 2 Multiply by 4: 4 x 56 = 224 3 Divide the product by 7: 224/7 = 32 4 So 56 divided by 1 3/4 = 32. Notice that numbers divisible by 7 will produce whole- number quotients. For numbers not divisible by 7, your calculator will give you long decimal results that are not exact. 1 Select a 2-digit number. 2 Multiply it by 7. 3 Move the decimal point 1 place to the left.
- 57. Example: 1 The numberchosen to multiply by 1 3/7 is 36. 2 Multiply by 7: 7 x 36 = 210 + 42 = 252 3 Move the decimal point 1 place to the left: 25.2 4 So 36 divided by 1 3/7 = 25.2. See the pattern? 1 The number chosen to multiply by 1 3/7 is 51. 2 Multiply by 7: 7 x 51 = 357 3 Move the decimal point 1 place to the left: 35.7 4 So 36 divided by 1 3/7 = 35.7. 1 Select a 2- or 3-digit number. 2 Multiply by 4 (or by 2 twice). 3 Move the decimal point one place to the left. Example: 1 The 2-digit number chosen to divide by 2 1/2 is 86. 2 Multiply by 4: 4 x 80 + 4 x 6 = 320 + 24 = 344 3 Move the decimal point one place to the left: 34.4 4 So 86 divided by 2 1/2 = 34.4. See the pattern? 1 The 3-digit number chosen to divide by 2 1/2 is 624. 2 Multiply by 2: 2 x 624 = 1248 3 Multiply by 2: 2 x 1248 = 2400 + 96 = 2496 4 Move the decimal point one place to the left: 249.6 5 So 624 divided by 2 1/2 = 249.6. Multiply by 4 when this is easy; otherwise use two steps and multiply by 2 twice. Dividing a 2-digit number by 2 1/3 1 Select a 2-digit number. 2 Multiply by 3. 3 Divide the result by 7. Example: 1 he 2-digit number chosen to divide by 2 1/3 is 42. 2 Multiply by 3:
- 58. 3 x 42= 126 3 Divide by 7: 126/7 = 18 4 So 42 divided by 2 1/3 = 18. See the pattern? 1 The 2-digit number chosen to divide by 2 1/3 is 73. 2 Multiply by 3: 3 x 73 = 219 3 Divide by 7: 219/7 = 31 2/7 4 So 73 divided by 2 1/3 = 31 2/7. If the number chosen is divisible by 7, the quotient will be a whole number. If the number is not divisible by 7, a calculator user will get a long, inexact decimal, while your answer will be exact. 1 Select a 2-digit number. 2 Multiply by 3. 3 Divide by 8. Example: 1 The 2-digit number chosen to divide by 2 2/3 is 32. 2 Multiply by 3: 3 x 32 = 96 3 Divide by 8: 96/8 = 12 4 So 32 divided by 2 2/3 = 12. See the pattern? 1 The 2-digit number chosen to divide by 2 2/3 is 61. 2 Multiply by 3: 3 x 61 = 183 3 Divide by 8: 183/8 = 22 7/8 4 So 61 divided by 2 2/3 = 22 7/8. Using this method, your answers will be exact. Those using calculators will only get approximations. 1 Select a 2-digit number. 2 Mltmultiiply the number by 2. 3 Divide the product by 7. Example: 1 The 2-digit number chosen to divide by 3 1/2 is 42. 2 Multiply by 2: 2 x 42 = 84 3 Divide by 7: 84/7 = 12 4 So 42 divided by 3 1/2 = 12. See the pattern? 1 The 2-digit number chosen to divide by 3 1/2 is 61. 2 Multiply by 2:
- 59. 2 x 61= 122 3 Divide by 7: 122/7 = 17 3/7 4 So 61 divided by 3 1/2 = 17 3/7. If the number chosen is divisible by 7, the answer will be a whole number. For numbers not divisible by 7, a calculator will get a repeating decimal, but your fractional answer will be exact. 1 Select a 2- or 3-digit number. 2 Multiply by 3. 3 Move the decimal point one place to the left. Example: 1 The 2-digit number chosen to divide by 3 1/3 is 72. 2 Multiply by 3: 72 x 3 = 216 3 Move the decimal point one place to the left: 21.6 4 So 72 divided by 3 1/3 = 21.6. See the pattern? 1 The 2-digit number chosen to divide by 3 1/3 is 48. 2 Multiply by 3: 48 x 3 = 120 + 24 = 144 3 Move the decimal point one place to the left: 14.4 4 So 48 divided by 3 1/3 = 14.4. After practicing, choose larger numbers. Insist on exact answers (you won't get an exact answer if you divide by 3.3 using a calculator). Multiply from left to right in steps and impress your friends with your mental powers. Dividing a 2-digit number by 375 1 Select a 2-digit number. 2 Multiply by 8. 3 Divide the product by 3 (express remainder as a fraction). 4 Move the decimal point three places to the left. Example:
- 60. 1 The numberchosen to divide by 375 is 32. 2 Multiply by 8: 8 x 32 = 240 + 16 = 256 3 Divide by 3: 256/3 = 85.3 1/3 4 Move the decimal point 3 places to the left: .0853 1/3 5 So 32 divided by 375 = .0853 1/3. See the pattern? 1 The number chosen to divide by 375 is 61. 2 Multiply by 8: 8 x 61 = 480 + 8 = 488 3 Divide by 3: 488/3 = 162 2/3 4 Move the decimal point 3 places to the left: .162 2/3 5 So 61 divided by 375 = .162 2/3. 72 3 Divide by 9: 72/9 = 8 4 So 36 divided by 4 1/2 = 8. For numbers not divisible by 9, your calculator will get a repeating decimal, but your fractional answer will be exact. Dividing a 2- digit number by 625 1 Select a 2-digit number. 2 Multiply by 8. 3 Divide the product by 5. 4 Move the decimal point 3 places to the left. Example: 1 The 2-digit number chosen to divide by 625 is 65. 2 Multiply by 8: 8 x 65 = 480 + 40 = 520 3 Divide by 5: 520/5 = 104 4 Move the decimal point 3 places to the left: .104 5 So 65 divided by 625 = .104. See the pattern?
- 61. 1 The 2-digitnumber chosen to divide by 625 is 32. 2 Multiply by 8: 8 x 32 = 240 + 16 = 256 3 Divide by 5: 256/5 = 51.2 4 Move the decimal point 3 places to the left: .0512 5 So 32 divided by 625 = .0512. Dividing a 2-digit number by 7 1/2 1 Select a 2-digit number. 2 Multiply it by 4 (or by 2 twice). 3 Divide by 3. 4 Move the decimal point 1 place to the left. 2M u lt i p l Example: y b 1 The 2-digit number chosen to multiply by 7 1/2 is 42. y 4: 42 x 4 = 168 3 Divide by 3: 168/3 = 56 4 Move the decimal point 1 place to the left: 5.6 5 So 42 divided by 7 1/2 = 5.6. See the pattern? 1 The 2-digit number chosen to multiply by 7 1/2 is 93. 2 Multiply by Dividing a 2-digit number by 4 1/2 2 1 Select a 2-digit number. 2 Multiply the number by 2. 3 Divide the product by 9. Example: 1 The 2-digit number chosen to divide by 4 1/2 is 62. 2 3 Multiply by 2: 2 x 62 = 124 Divide by 9: 124/9 = 13 7/9 4 So 62 divideby 4 1/2 = 13 7/9. See the pattern? 1 The 2-digit number chosen to divide by 4 1/2 is 36. 4: 93 x 4 = 360 + 12 = 372 3 Divide by 3: 372/3 = 124 4 Move the
- 62. decimal point 1place to the left: 12.4 5 So 93 divided by 7 1/2 = 12.4. Practice and you will soon be cranking out these quotients with speed and accuracy. Dividing a 2- or 3-digit number by 16 2/3 1 Select a 2-digit number. (Choose larger numbers when you feel sure about the method.) 2 Multiply by 6 (or by 3 and then 2). 3 Move the decimal point two places to the left. Example: 1 The 2-digit number chosen to divide by 16 2/3 is 72. 2 Multiply by 3: 3 x 72 = 216 3 Multiply by 2: 2 x 216 = 432 4 Move the decimal point 2 places to the left: 4.32 5 So 72 divided by 16 2/3 = 4.32. See the pattern? 1 The 2-digit number chosen to divide by 16 2/3 is 212. 2 Multiply by 3: 3 x 212 = 636 3 Multiply by 2: 2 x 636 = 1200 + 72 = 1272 4 Move the decimal point 2 places to the left: 12.72 5 So 212 divided by 16 2/3 = 12.72. Practice multiplying by 3, then by 2, and you will be able to do these problems quickly. Divisibility Rules Why do these 'rules' work? - Dr. Rob Divisibilidad por 13 y por números primos (13,17,19...) -en español, de la lista SNARK From the Archives of the Math Forum's Internet project Ask Dr. Mathour thanks to Ethan 'Dr. Math' Magness, Steven 'Dr. Math' Sinnott,
- 63. nd, for theexplanation of why these rules work, Robert L. Ward (Dr. Rob). Dividing by 3 Add up the digits: if the sum is divisible by three, then the number is as well. Examples: 1 111111: the digits add to 6 so the whole number is divisible by three. 2 87687687. The digits add up to 57, and 5 plus seven is 12, so the original number is divisible by three. Why does the 'divisibility by 3' rule work? From: "Dr. Math" To: keving@ecentral.com (Kevin Gallagher) Subject: Re: Divisibility of a number by 3 As Kevin Gallagher wrote to Dr. Math On 5/11/96 at 21:35:40 (Eastern Time), >I'm looking for a SIMPLE way to explain to several very bright 2nd >graders why the divisibility by 3 rule works, i.e. add up all the >digits; if the sum is evenly divisible by 3, then the number is as well. >Thanks! >Kevin Gallagher The only way that I can think of to explain this would be as follows: Look at a 2 digit number: 10a+b=9a+(a+b). We know that 9a is divisible by 3, so 10a+b will be divisible by 3 if and only if a+b is. Similarly, 100a+10b+c=99a+9b+(a+b+c), and 99a+9b is divisible by 3, so the total will be iff a+b+c is. This explanation also works to prove the divisibility by 9 test. It clearly originates from modular arithmetic ideas, and I'm not sure
- 64. if it's simpleenough, but it's the only explanation I can think of. Doctor Darren, The Math Forum Check out our web site -http://forum.swarthmore.edu/dr.math/ Dividing by 4 ook at the last two digits. If they are divisible by 4, the number is as well. Examples: 1 100 is divisible by 4. 2 1732782989264864826421834612 is divisible by four also, because 12 is divisible by four. Dividing by 5 If the last digit is a five or a zero, then the number is divisible by 5. Dividing by 6 Check 3 and 2. If the number is divisible by both 3 and 2, it is divisible by 6 as well. Robert Rusher writes in: Another easy way to tell if a [multi-digit] number is divisible by six . . . is to look at its [ones digit]: if it is even, and the sum of the [digits] is a multiple of 3, then the number is divisible by 6. Dividing by 7 To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number. Example: If you had 203, you would double the last digit to get six, and ubtract that from 20 to get 14. If you get an answer divisible by 7 (including zero), then the original number is divisible by seven. If you don't know the new number's divisibility, you can apply the rule again. Dividing by 8
- 65. Check the lastthree digits. Since 1000 is divisible by 8, if the last three digits of a number are divisible by 8, then so is the whole number. Example: 33333888is divisible by 8; 33333886 isn't. How can you tell whether the last three digits are divisible by 8? Phillip McReynolds answers: If the first digit is even, the number is divisible by 8 if the last two digits are. If the first digit is odd, subtract 4 from the last two digits; the number will be divisible by 8 if the r esulting last two digits are. So, to continue the last example, 33333888 is divisible by 8 because the digit in the hundreds place is an even number, and the last two digits are 88, which is divisible by 8. 33333886 is not divisible by 8 because the digit in the hundreds place is an even number, but the last two digits are 86, which is not divisible by 8. Dividing by 9 Add the digits. If they are divisible by nine, then the number is as well. This holds for any power of three. Dividing by 10 If the number ends in 0, it is divisible by 10. Dividing by 11Let's look at 352, which is divisible by 11; the answer is 32. 3+2 is 5; another way to say this is that 35 -2 is 33. Now look at 3531, which is also divisible by 11. It is not a coincidence that 353-1 is 352 and 11 x 321 is 3531. Here is a generalization of this system. Let's look at the number 94186565. First we want to find whether it is divisible by 11, but on the way we are going to save the numbers that we use: in every step we will subtract the last digit from the other digits, then save the subtracted amount in order. Start with 9418656 - 5 = 9418651 SAVE 5 Then 941865 - 1 = 941864 SAVE 1 Then 94186 - 4 = 94182 SAVE 4 Then 9418 - 2 = 9416 SAVE 2 Then 941 - 6 = 935
- 66. SAVE 6 Then93 - 5 = 88 SAVE 5 Then 8 - 8 = 0 SAVE 8 Now write the numbers we saved in reverse order, and we have 8562415, which multiplied by 11 is 94186565. Here's an even easier method, contributed by Chis Foren:Take any number, such as 365167484.Add the 1,3,5,7,..,digits.... .3 + 5 + 6 + 4 + 4 = 22Add the 2,4,6,8,..,digits.....6 + 1 + 7 + 8 = 22If the difference, including 0, is divisible by 11, then so is the number. 22 - 22 = 0 so 365167484 is evenly divisible by 11.See also Divisibility by 11 in the Dr. Math archives. Dividing by 12 Check for divisibility by 3 and 4. Dividing by 13 Here's a straightforward method supplied by Scott Fellows: D e lete the last digit from the given number. Then subtract nine times the deleted digit from the remaining number. If what is left is divisible by 13, then so is the original number. And here's a more complex method that can be extended to other formulas: 1 = 1 (mod 13)10 = -3 (mod 13) (i.e., 10 - -3 is divisible by 13)100 = -4 (mod 13) (i.e., 100 - -4 is divisible by 13)1000 = -1 (mod 13) (i.e., 1000 - -1 is divisible by 13)10000 = 3 (mod 13)100000 = 4 (mod 13)1000000 = 1 (mod 13) Call the ones digit a, the tens digit b, the hundreds digit c, ..... and you get: a - 3*b - 4*c - d + 3*e + 4*f + g - .....
- 67. If this numberis divisible by 13, then so is the original number. You can keep using this technique to get other formulas for divisibility for prime numbers. For composite numbers just check for divisibility by divisors. Finding 2 1/2 percent of a number 1 Choose a number (start with 2 digits and advance to 3 with practice). 2 Divide by 4 (or divide twice by 2). 3 Move the decimal point one place to the left. Example: 1 If the number selected is 86: 2 Divide 86 by 4: 86/4 = 21.5 3 Move the decimal point one place to the left.: 2.15 4 So 2 1/2% of 86 = 2.15. See the pattern? 1 If the number selected is 648: 2 Divide 648 by 2 twice: 648/2 = 324, 324/2 = 162 3 Move the decimal point one place to the left.: 16.2 4 So 2 1/2% of 648 = 16.2. Practice dividing by 4, or by 2 twice, and you will be able to find these answers faster than with a calculator. Finding 5 percent of a number 1 Choose a large number (or sum of money). 2 Move the decimal point one place to the left. 3 Divide by 2 (take half of it). Example: 1 If the amount of money selected is $850: 2 Move the decimal point one place to the left.: 85 3 Divide by 2: 85/2 = 42.50 4 So 5% of $850 = $42.50. See the pattern?
- 68. 1 If theamount of money selected is $4500: 2 Move the decimal point one place to the left.: 450 3 Divide by 2: 450/2 = 225 4 So 5% of $4500 = $225. Finding 15 percent of a number 1 Choose a 2-digit number. 2 Multiply the number by 3. 3 Divide by 2 . 4 Move the decimal point one place to the left. Example: 1 If the number selected is 43: 2 Multiply by 3: 3 x 43 = 129 3 Divide by 2: 129/2 = 64.5 4 Move the decimal point one place to the left: 6.45 5 So 15% of 43 = 6.45. See the pattern? 1 If the number selected is 72: 2 Multiply by 3: 3 x 72 = 216 3 Divide by 2: 216/2 = 108 4 Move the decimal point one place to the left: 10.8 5S o 15% of72=10.8 . Finding 45 percent of a number 1 Choose a 2-digit number. 2 Multiply the number by 9. 3 Divide by 2. 4 Move the decimal point one place to the left. Example: 1 If the number selected is 36: 2 Multiply by 9: 9 x 36 = 270 + 54 = 324 3 Divide by 2: 324/2 = 163 4 Move the decimal point one place to the left : 16.2 5 So 45% of 36 = 16.2. See the pattern?
- 69. If the numberselected is 52: Multiply by 9: 9 x 52 = 450 + 18 = 468 Divide by 2: 468/2 = 234 Move the decimal point one place to the left: 23.4 So 45% of 52 = 23.4. Finding 55 percent of a number n 1 Choose a 2-digit number. e 2 Multiply the number by 11. (Add digits from right to left - see examples). x 3 Divide by 2. t 4 Move the decimal point one place to the left. d i Example: g it 1 If the number selected is 81: t 2 Multiply by 11: 11 x 81 = 891 o right digit is 1 l eft is 1 + 8 = 9 last digit to left is 8 3 Divide by 2: 891/2 = 445.5 4 Move the decimal point one place to the left: 44.55 5 So 55% of 81 = 44.55. See the pattern? 1 If the number selected is 59: 2. Multiply by 11: 11 x 59 = 649 right digit is 9 next digit to left is 9 + 5 = 14 (use the 4 and carry 1) last digit to left is 5 + 1 = 6 2 Divide by 2: 649/2 = 324.5 3 Move the decimal point one place to the left: 32.45 4 So 55% of 59 = 32.45. Why do the divisibility "rules" work? The Power of Modulo Arithmetic Modulo arithmetic is a powerful tool that can be used to test for divisibility by any number. The great disadvantage of using modulo arithmetic to test for divisibility is the fact that it is usually a slow method. Under some circumstances, however, the
- 70. application of moduloarithmetic leads to divisibility rules that can be used. These rules are important to mathematics because they save us lots of time and effort. In fact, many of the divisibility rules that are commonly used (rules for 3, 9, 11) have their roots in modulo arithmetic. This page will show you how some of these common divisibility rules are connected to modulo arithmetic. Applying the Rules of Modulo Arithmetic The rules of modulo arithmetic state that the number N is divisible by some number (P) if the above expression is also divisible by P after the base (10) is replaced by the remainder of 10 divided by P. In compact notation, this remainder is denoted by (10 mod P). T hus, N is divisible by P if the following expression is also divisible by P: [Dn * (10 mod P)^(n-1)] + ... + [D2 * (10 mod P)^1] + [D1 * (10 mod P)^0]. Origin of Divisibility Rules for 3, 9, and 11 D We can now see how the divisibility rules for P = 3, 9, and 11 are rooted in modulo arithmetic. n Consider the case where P = 3. . Because 10 mod 3 is equal to 1, any number is divisible by 3 if the following expression is . also divisible by 3: . - [Dn * (1)^n-1] + ... + [D2 * (1)^1] + [D1* D 2 (1)^0]. Because 1 raised to any power is equal to 1, the above
- 71. + expression can besimplified as: Dn + ... + D2 + D1 + D0, D which is equal to the sum of digits in the number. Thus modulo arithmetic allows us to 1 state that any number is divisible by 3 if the sum of its digits is Because 10 mod 9 is also 1, any number is divisible by 9 is the sum of its digits are also divisible by 9. Thus, the divisibility rules for 3 and 9 come directly from modulo n arithmetic. n) Finally, consider the case where P = 11. a Because 10 mod 11 is equal to -1, any number is divisible by 11 if the n following expression is also divisible by 11: d D[Dn * (-1)^n-1] + ... + [D2 * (-1)^1] + [D1* (-1)^0]. n Because -1 raised to any even power is equal to 1 and -1 raised to any odd +power is equal to -1, the above expression becomes a summation of the digits involving alternated signs: . . . - D2 + D1 (for odd n). This "alternating" summation rule for P = 11 is well known and has two versions. The first version states a number is divisible by 11 if the alternating sum of the digits is also divisilbe by 11 (i.e., the alternating sum is a muliple of 11). The second version states that a number is divisible by 11 if the sum of every other digit starting with the rightmost digit is equal to the sum of
- 72. every other digitstarting with the second digit from the right (i.e., the alternating sum is 0). Both of these versions come directly from the rules of modulo arithmetic. Bases Other Than 10 v Up to this point, we have only considered numbers written in base 10. A number can, however, be i written in any base (B). Such a number can be expressed by the following summation: g it N = [Dn * B^n-1] + [Dn-1 * B^(n-2)] + ... + [D2 * B^1] + [D1* B^0]. s " The rules of modulo arithmetic apply no matter what base is used . These rules tell us that N is divisible by P if the following expression is also divisible by P: [Dn * (B mod P)^n-1] + ... + [D2 * (B mod P)^1] + [D1* (B mod P)^0]. Most bases other than 10 are difficult to use, but bases that are powers of 10 can easily be used to construct divisibility rules using modulo arithmetic. Consider writing the base-10 number N = 1233457 in base- 100. This can be done by starting with the rightmost digit and grouping the digits in pairs of two. bEach grouping of two digits is considered a single "digit" when the number is ywritten in base-100. Written in base-100, the number is 1 23 34 57 where 57, 34, 123, and 1 are considered single "digits." In a similar fashion, this number written 1in base-1000 is 1 233 457 where 457, 233, and 1 are considered single "digits." .Once the base-10 digits are grouped to form the "digits," the above expression Bcan be used to test for divisibility. Consider P = 7. Use the base, B = 1000. Because 1000 mod 7 = -1, the
- 73. alternating summation rulea (used for P = 11 in base-10) can be applied. Before applying this rule, the base-10 pdigits must be properly grouped to form base-1000 "digits." p e Stryker explains it this way: When you are multiplying by 9, on your fingers (starting with your thumb) count the number you are multiplying by and hold down that finger. The number of fingers before the finger held down is the first digit of the answer and the number of finger after the finger held down is the second digit of the answer. Example: 2 x 9. your index finder is held down, your thumb is before, representing 1, and there are eight fingers after your index finger, representing 18. ● Polly Norris suggests: When you multiply a number times 9, count back one from that number to get the beginning of your product. ( 5 x 9: one less than 5 is 4). To get the rest of your answer, just think of the add fact families for 9: 1+8=92+7=93+6=94+5=9 8+1=97+2=96+3=95+4=9 5 x 9 = 4_. Just think to yourself: 4 + _ = 9 because the digits in your product always add up to 9 when one of the factors is 9. Therefore, 4 + 5 = 9 and your answer is 45! I use this method to teach the "nines in multiplication to my third graders and they learn them in one lesson!
- 74. Tamzo explains thisa little differently: 1 Take the number you are multiplying 9 by and subtract one. That number is the first number in the solution. 2 Then subtract that number from nine. That number is the second number of the solution. Examples: 4 * 9 = 36 1 4-1=3 2 9-3=6 3 solution = 36 8 * 9 = 72 1 8-1=7 2 9-7=2 3 solution = 72 5 * 9 = 45 1 5-1=4 2 9-4=5 3 solution = 45 ● Sergey writes in: Take the one-digit number you are multipling by nine, and insert a zero to its right. Then subtract the original number from it. For example: if the problem is 9 * 6, insert a zero to the right of the six, then subtract six: 9 * 6 = 60 - 6 = 54 Multiplying a 2-digit number by 11 ● A tip sent in by Bill Eldridge: Simply add the first and second digits and place the result between them. Here's an example using 24 as the 2-digit number to be multiplied by 11: 2 + 4 = 6 so 24 x 11 = 264. This can be done using any 2-digit number. (If the sum is 10 or greater , don't forget to carry the one.) Multiplying any number by 11
- 75. ● Lonnie DennisII writes in: Let's say, for example, you wanted to multiply 54321 by 11. First, let's look at the problem the long way... 54321 x 11 54321 + 543210 = 597531 Now let's look at the easy way... 11 x 54321 4432 =51 +5 +3 +2 +1 531 Do you see the pattern? In a way, you're simply adding the digit to whatever comes before it. But you must work from right to left. The reason I work from right to left is that if the numbers, when added together, sum to more than 9, then you have something to carry over. Let's look at another example... 11 x 9527136 Well, we know that 6 will be the last number in the answer. So the answer now is ???????6. Calculate the tens place: 6+3=9, so now we know that the product has the form ??????96. 3+1=4, so now we know that the
- 76. product has theform ?????496. 1+7=8, so ????8496. 7+2=9, so ???98496. 2+5=7, so ??798496. 5+9=14. Here's where carrying digits comes in: we fill in the hundred thousands place with the ones digit of the sum 5+9, and our product has the form ?4798496. We will carry the extra 10 over to the next (and final) place. 9+0=9, but we need to add the one carried from the previous sum: 9+0+1=10. So the product is 104798496. Calculation Tips & Tricks Multiplication Tips Multiplying by five ● Jenny Logwood writes: Here is an easy way to find an answer to a 5 times question. If you are multiplying 5 times an even number: halve the number you are multiplying by and place a zero after the number. Example: 5 x 6, half of 6 is 3, add a zero for an answer of 30. Another example: 5 x 8, half of 8 is 4, add a zero for an answer of 40. If you are multiplying 5 times an odd number: subtract one from the number you are multiplying, then halve that number and place a 5 after the resulting number. Example: 5 x 7: -1 from 7 is 6, half of 6 is 3, place a 5 at the end of the resulting number to produce the
- 77. number 35. Anotherexample: 5 x 3: -1 from 3 is 2, half of 2 is 1, place a 5 at the end of this number to produce 15. ● Doug Elliott adds: To square a number that ends in 5, multiply the tens digit by (itself+1), then append 25. For example: to calculate 25 x 25, first do 2 x 3 = 6, then append 25 to this result; the answer is 625. Other examples: 55 x 55; 5 x 6 = 30, answer is 3025. You can also square three digit numbers this way, by starting with the the first two digits: 995 x 995; 99 x 100 = 9900, answer is 990025. Multiplying by nine � Diana Grinwis says: To multiply by nine on your fingers, hold up ten fingers - if the problem is 9 x 8 you just put down your 8 finger and there's your answer: 72. (If the problem is 9 x 7 just put down your 7 finger: 63.) � Laurie Stryker explains it this way: When you are multiplying by 9, on your fingers (starting with your thumb) count the number you are multiplying by and hold down that finger. The number of fingers before the finger held down is the first digit of the answer and the number of finger after the finger held down is the second digit of the answer. Example: 2 x 9. your index finder is held down, your thumb is
- 78. before, representing 1,and there are eight fingers after your index finger, representing 18. Norris suggests: When you multiply a number times 9, count back one from that H number to get the beginning of your product. (5 x 9: one less than 5 is 4). e r To get the rest of your answer, just think of the add fact families for 9: e ' 1 + 8 = 9 2 + 7 = 9 3 + 6 = 9 4 + 5 = 9 s8 + 1 = 9 7 + 2 = 9 6 + 3 = 9 5+4=9 5 x 9 = 4_. Just think to yourself: 4 + _ = 9 because the digits in your product n always add up to 9 when one of the factors is 9. Therefore, 4 + 5 = 9 and your answer is 45! I use this method to each the "nines" in multiplication to my third e graders and they learn them in one lesson! Tamzo explains this a little differently: 1. Take the number you are multiplying 9 by and subtract one. That number is the first e number in the solution. 2 Then subtract t that number from nine. That number is the second number of the solution. Examples: n 4 * 9 = 36 g1 4-1=3 2 9-3=6 2 3 solution = 36 4 8 * 9 = 72 a s 1 8-1=7 2 9-7=2 t 3 solution = 72 h
- 79. e 5 * 9= 45 2 -d 1 5-1=4 i 2 9-4=5 g3 solution = 45 i t Multiplying a 2-digit number by 11 n ● A tip sent in by Bill Eldridge: Simply add the first and second digits and place the result u between them. m b er to be multiplied by 11: 2 + 4 = 6 so 24 x 11 = 264. This can be done using any 2-digit number. (If the sum is 10 or g reater, don't forget to carry the one.) Multiplying any number by 11 ● Lonnie Dennis II writes in: Let's say, for example, you wanted to multiply 54321 by 11. First, let's look at the problem the long way... 54321 x 11 54321 + 543210 = 597531 Now let's look at the easy way... 11 x 54321 =51 +5 +3 +2 +1 Do you see the pattern? In a way, you're simply adding the digit to whatever comes before it. But you must work from right to left. The reason I work from right to left is that if the numbers, when added together, sum to more than 9, then you have something to carry over.
- 80. Let's look atanother example... 11 x 9527136 Well, we know that 6 will be the last number in the answer. So the answer now is 6. Calculate the tens place: 6+3=9, so now we know that the product has the form ??????96. 3+1=4, so now we know that the product has the form?????496. 1+7=8, so ????8496. 7+2=9, so ???98496. 2+5=7, so ??798496. 5+9=14. Here's where c arrying digits comes in: we fill in the hundred thousands place with the ones digit of the sum 5+9, and our product has the form ? 4798496. We will carry the extra 10 over to the next (and final) place. 9+0=9, but we need to add the one carried from the previous sum: 9+0+1=10. So the product is 104798496. Multiplying a 3-digit number by 99 1 Select a 3-digit number. 2 Subtract the 1st digit plus 1 from the number. X X X _ _ 3 Subtract the last two digits of the number from 100. _ _ _ X X Example: 1 The 3-digit number chosen to multiply by 99 is 274. 2 Subtract the 1st digit + 1 from the number: 274 - 3 = 271 : 2 7 1 _ _ 3 Subtract the last two digits from 100: 100 - 74 = 26: _ _ _ 2 6. 4 So 274 x 99 = 27126. See the pattern? 1 The 3-digit number chosen to multiply by 99 is 924. 2 Subtrac t the 1st digit + 1 from the number: 924 - 10 = 914 : 9 1 4 _ _ 3 Subtract the last two digits from 100:
- 81. 100 - 24= 76: _ _ _ 7 6. 4. So 924 x 99 = 91476. Multiplying a 3-digit number by 143 1 Select a 3-digit number. 2 Repeat the 3 digits to produce a 6-digit number. 3 Divide by 7. Example: 1 The 3-digit number chosen to multiply by 143 is 123. 2 Repeat 3 digits: 123123 3 Divide by 7: 123123/7 = 17589 4 So 143 x 123 = 17,589. See the pattern? 1 The 3-digit number chosen to multiply by 143 is 765. 2R e p e a t 3 d i g it s : 7 6 5765 3 Divide by 7: 765765/7 = 109395 4 So 143 x 765 = 109,395. Multiplying a 4-digit number by 99 1 Select a 4-digit number. 2 Subtract the first two digits plus 1 from the number. X X X X _ _ 3 Subtract the last two digits of the number from 100. ____XX Easier example: 1 Choose a 4-digit with digits from smaller to larger: 2368. 2 The first two digits will be the same: 2 3 _ _ _ _. 3 Subtract the first two digits + 1 from the last two digits: 23 + 1 = 24, 68 - 24 = 44: _ _ 4 4 4 Subtract the last two digits from 100: 100 - 68 = 32: _ _ _ _ 3 2 5 So 2368 x 99 = 234432. See the pattern? Advanced example: 1 The 4-digit number chosen to multiply by 99 is 3512. 2. Subtract the first two digits + 1 from the number:
- 82. 35 + 1= 36, 3512 - 36 = 3512 - 30 - 6 = 3482 - 6 = 3476: 3 4 7 6 __ 2 Subtract the last two digits from 100: 100 - 12 = 88: _ _ _ _ 8 8 3 So 3512 x 99 = 347688. Start out with easier examples and graduate to more difficult ones. Remember to subtract from left to right in increments. Multiplying a repeating 9-digit number by 3, then dividing by 12345679 1 Choose a repeating 9-digit number (one whose numbers are all the same). 2 Multiply one digit by 9. 3 Multiply by 3. Example: 1 If the first number is 333333333: 2 Multiply one digit by 9: 3 x 9 = 27. 3 Multiply 3: 27 x 3 = 81. 4 So 333333333 x 3 / 12345679 = 81. See the pattern? 1 If the first number is 666666666: 2 Multiply one digit by 9: 9 x 6 = 54. 3 Multiply 3: 54 x 3 = 162. 4 So 666666666 x 3 / 12345679 = 162. Change the second multiplier to other numbers and create many variations of this exercise. Multiplying two mixed numbers 1 Select a mixed number. (1 2/3, 5 1/8, etc.) 2 Select a multiplier using the same whole number and a fraction that sums to 1 with the first number's fractional part. 3 The whole number in the answer will be the whole number times the next number. 4 To find the fraction, multiply the two fractional parts. Example:
- 83. 1 The firstmixed number chosen is 7 1/4. 2 Select 7 3/4 as the multiplier (same whole number, fraction that sums to 1 with the first fraction. 3 The product's whole number will be 7 x 8 (next number) = 56. 4 The product's fraction will be 1/4 x 3/4 = 3/16. 5 So 7 1/4 x 7 3/4 = 56 3/16. See the pattern? 1 The first mixed number chosen is 8 1/3. f 2 Select 8 2/3 as the multiplier (same whole number, r action that sums to 1 with the first fraction. 5 3 The product's whole number will be 8 x 9 (next number) = 72. 4 The product's fraction will be 1/3 x 2/3 = 2/9. 6 So 8 1/3 x 8 2/3 = 72 2/9. One more: 1 The first mixed number chosen is 20 1/8. 2 Select 20 7/8 as the multiplier (same whole number, fraction that sums to 1 with the first fraction. 3 The product's whole number will be 20 x 21 (next number) = 420. 4 The product's fraction will be 1/8 x 7/8 = 7/64. 5 So 20 1/8 x 20 7/8 = 420 7/64. Multiplying a 4-digit number by 137, then by 73; subtract 1200 1 Choose a 4-digit number (one whose numbers are all the same). 2 Multiply by 137 and multiply the product by 73. 3 The second
- 84. product is an8-digit number: the original number and a repeat of the digits. 4 Subtract 1200. Example: 1 If the first number is 6241: 2 The answer to the two multiplications is 62416241 3 Subtract 1200: 62416241 - 1200 = 62415041 4 So 6241 x 137 x 73 = 62,415,041. See the pattern? 1 If the first number is 7834: 2 The answer to the two multiplications is 78347834 3 Subtract 1200: 78347834 - 1200 = 78346634 4 So 7834 x 137 x 73 = 78,346,634. You can expand this exercise by changing the final step to another calculation: add or subtract a number that is easy to handle. In this way you can create many new Multiplying a 5-digit number by 9091, then by 11; subtract 1200 act 1200. Exam ple: 1 If the first number is 34986: 2 The answer to the two multiplications is 3498634986 3 Subtract 1200: 3498634986 - 1200 = 3498633786 4 So 34986 x 9091 x 11 = 3,498,633,786. See the pattern? 1 If the first number is 22854: 2 The answer to the two multiplications i
- 85. 2285422854 3 Subtract1200: 2285422854 - 1200 = 2285421654 4 So 22854 x 9091 x 11 = 2,285,421,654. You can expand this exercise by changing the final step to another calculation: add or subtract a number that is easy to handle. In this way you can create many new Multiplying a multi-digit sequence of numbers by 8 and adding the last digit 1 Choose a number made up of consecutive digits beginning with 1 (123, 1234 etc.). 2 Multiply it by 8 and add the last digit of the original number. 3 The answer will have the same number of digits as the first number selected; the digits will be decreasing d igits starting with 9. Example: 1 If the number selected is 12345: 2 Multiply by 8, add 5: 3 The answer is 98765 (5 digits). 4 So 12345 multiplied by 8 plus 5 is 98765. See the pattern? 1 If the number selected is 12345678: 2 Multiply by 8, add 8: 3 The answer is 98765432 (8 digits). 4 So 12345678 multiplied by 8 plus 8 is 98765432. More than one example will probably give away the secret, but this ought to be good for one-shot tries. Multiply a multi-digit sequence of numbers 1 Choose a 5-digit number. 2 Multiply by 9091 and multiply the product by 11.
- 86. 3 The secondproduct is a 10-digit number: the original number and a repeat of the digits. by 8, add a number, subtract 20 1 Choose a number made up of consecutive digits beginning with 1 (123, 1234 etc. - up to 9 digits). 2 Multiply it by 8. 3 Add the last digit of the original sequence. The answer will be a declining sequence beginning with 9 and the same length as the original sequence. 4 Subtract 20. Example: 1 If the number selected is 123: 2 Multiply by 8, add 3 (descending sequence beginning with 9 and 3 digits long): 987 3 Subtract 20: 987 - 20 = 967 4 So 123 multiplied by 8 plus 3 - 20 is 967. See the pattern? 1 If the number selected is 123456: 2 The answer before the last subtraction is 987654 (descending sequence from 9 with 6 digits) 3 Subtract 20: 987654 - 20 = 987634 4 So 123456 multiplied by 8 + 6 - 20 is 987634. Extend the number of examples by changing the last step – or subtract or add different numbers. Multiply the sum and difference of a 2-and a 1-digit number re nc e. Ex a
- 87. m ple: 1 If thenumbers selected are 80 and 3: 2 Add the numbers: 80 + 3 = 83. 3 Subtract 3 from 80: 80 - 3 = 77. 4 80 x 80 - 3 x 3 = 6400 -9= 6391. 5 So 83 x 77 = 6391. See the pattern? 1 If the numbers selected are 60 and 8: 2 Add the numbers: 60 + 8 = 68. 3 Subtract 8 from 60: 60 - 8 = 52. 4 60 x 60 - 8 x 8 = 3600 - 64 = 3536. So 68 x 52 = 3536. 1 Choose a two-digit multiple of 10 and a single-digit number (1- 9). d 2 Add the two numbers if 3 Find their difference (subtract the smaller from the larger). f 4 Multiply the sum and the difference by squaring the two numbers and finding the e Multiplying a 2-digit number by 1 1/6 1 Select a 2-digit number. 2 Multiply by 7. 3 Divide the result by 6. Example: 1 The 2-digit number chosen to multiply by 1 1/6 is 34. 2 Multiply by 7: 7 x 34 = 210 + 28 = 238 3 Divide by 6: 238/6 = 39 4/6 4 So 34 x1 1/6 = 39 2/3. See the pattern? 1 If the 2-digit number chosen to multiply by 1 1/6 is 57: 2 Multiply by 7: 7 x 57 = 350 + 49 = 399 3 Divide by 6: 399/6 = 66 3/6 4 So 57 x
- 88. 1 1/6 =66 1/2. Multiplying a 2-digit number by 1 1/8 2D 1 Multiply the number by 9. i vide by 8. Example: 1 If the 2-digit number chosen to multiply by 1 1/8 is 32: 2 Multiply by 9: 9 x 32 = 270 + 18 = 288 3 Divide by 8: 288/8 = 36 4 So 32 x 1 1/8 = 36. See the pattern? 1 If the number to be multiplied by 1 1/8 is 71: 2 Multiply by 9: 9 x 71 = 639 3 Divide by 8: 639/8 = 79 7/8 4 So 71 x 1 1/8 = 79 7/8. Multiplying a 2-digit number by 1 1/9 1 Select a 2-digit number. 2 Add a zero to the number. 3 Divide the result by 9. Example: 1 The 2-digit number chosen to multiply by 1 1/9 is 32. 2 Add zero: 320 3 Divide by 9: 320/9 = 35 5/9 4 So 32 x 1 1/9 = 35 5/9. See the pattern? 1 If the 2-digit number chosen to multiply by 1 1/9 is 74: 2 Add zero: 740 3 Divide by 9: 740/9 = 82 2/9 4 So 74 x 1 1/9 = 82 2/9. Those using calculators may have difficulty entering 1 1/9, and their answers will be repeating decimals. Your answer will be exact. Multiplying a 2-digit number by 1 1/4 1 Add a zero to the number. 2 Divide by 8. Example: 1 If the 2-digit number chosen to multiply by 1 1/4 is 37: 2 Add a zero: 370 3 Divide by 8: 370/8 = 46 1/4 4 So 37 x 1 1/4 = 46 1/4. See the pattern?
- 89. 1 If thenumber to be multiplied by 1 1/4 is 72: 2 Add a zero: 720 3 Divide by 8: 720/8 = 90 4 So 72 x 1 1/4 = 90. Multiplying a 2-digit number by 1 5/6 1 Select a 2-digit number. 2 Multiply by 11. 3 Divide the result by 6. Example: 1 The 2-digit number chosen to multiply by 1 5/6 is 34. 2 Multiply by 1 1: 11 x 34 = 374 (Add the digits left to right. The last digit is 4. The next digit is 4 + 3 = 7. The first digit is 3.) 3 Divide by 6: 374/6 = 62 2/6 4 So 34 x 1 5/6 = 62 1/3. See the pattern? 1 If the 2-digit number chosen to multiply by 1 5/6 is 62: 2 Multiply by 11: 11 x 62 = 682 3 Divide by 6: 682/6 = 113 4/6 4 So 62 x 1 5/6 = 113 2/3. To produce these products promptly, practice multiplying by 11. Multiplying a 2-digit number by 1 3/4 1 Multiply the number by 7. 2 Divide by 4. Example: 1 If the number chosen to multiply by 1 3/4 is 42: 2 Multiply by 7: 7 x 42 = 280 + 14 = 294 3 Divide by 4: 294/4 = 73 2/4 4 So 42 x 1 3/4 = 73 1/2. See the pattern? 1 If the number to be multiplied by 1 3/4 is 29: 2 Multiply by 7: 7 x 29 = 140 + 63 = 203 3 Divide by 4: 203/4 = 50 3/4 4 So 29 x 1 3/4 = 50 3/4. Practice, and these products will be fast and accurate. Multiplying a repeating 3-digit number by 2, then dividing by 37
- 90. 1 Choose arepeating 3-digit number (one whose numbers are all the same). 2 Add its digits (or multiply one of them by 3). 3 Multiply by 2. Example: 1 If the first number is 555: 2 Add its digits (or multiply 5 x 3) = 15. 3 Multiply 15 x 2 = 30. 4 So 555 x 2 / 37 = 30. See the pattern? 1 If the first number is 888: 2 Add 8 + 8 + 8 or multiply 8 x 3 = 24 . 3 Multiply 24 x 2 = 48. 4 So 888 x 2 / 37 = 48. Multiplying a 2-digit number by 2 3/4 1M ultiply the number by 11. Add the digits from right to left. 2 Divide by 4. Example: 1 If the number chosen to multiply by 2 3/4 is 78: 2 Multiply by 11. Add digits from right to left: _ _ 8 8 + 7 = 15 7 + 1 (carry) = 8 78 * 11 = 858 3 Divide by 4: 858/4 = 214 1/2 4 So 78 x 2 3/4 = 214 1/2. See the pattern? 1 If the number to be multiplied by 2 3/4 is 27: 2 Multiply by 11: 11 x 27 = 297 3 Divide by 4: 297/4 = 74 1/4 4 So 27 x 2 3/4 = 74 1/4. Multiplying a repeating 3-digit number by 5, then dividing by 37 1 Choose a repeating 3-digit number (one whose numbers are all the same). 2 Add its digits (or multiply one of them by 3). 3 Multiply by 5. Example:
- 91. 1 If thefirst number is 444: 2 Add its digits (or multiply 4 x 3) = 12. 3 Multiply 12 x 5 = 60. 4 So 444 x 5 / 37 = 60. See the pattern? 1 If the first number is 777: 2 Add 7 + 7 + 7 or multiply 7 x 3 = 21. 3 Multiply 21 x 5 = 105. 4 So 777 x 5 / 37 = 105. Expand this exercise to others of your choosing by multiplying by other numbers. You can produce many new exercises in this manner. Multiplying a 2-digit number by 7 1/2 1 Multiply by 3. 2 Divide by 4. 3 Add a zero or move the decimal point one place to the right. Example: 1 If the 2-digit number chosen to multiply by 7 1/2 is 64: 2 Multiply by 3: 3 x 64 = 180 + 12 = 192 3 Divide by 4: 192/4 = 48 4 Add zero: 480 5 So 64 x 7 1/2 = 480. See the pattern? 1 If the number to be multiplied by 7 1/2 is 93: 2 Multiply by 3: 3 x 93 = 279. 3 Divide by 4: 279/4 = 69 3/4 or 69.75 4 Move the decimal point one place t o the right: 697.5 5 So 93 x 7 1/2 = 697.5 Multiplying a 2-digit number by 12 1/2 1 Select a 2-digit number . 2 Divide it by 8 (or by 2 three times). 3 Move the decimal point 2 places to the right. Example: 1 The 2-digit number chosen to multiply by 12 1/2 is 28. 2 Divide by 8: 28/8
- 92. = 3.5 3Move the decimal point 2 places to the right: 350 4 So 28 x 12 1/2 = 350. See the pattern? 1 If the 2-digit number chosen to multiply by 12 1/2 is 58: 2 Divide by 8 by 2 three times): 58/2 = 29 29/2 = 14.5 14.5/2 = 7.25 3M o v e t h e d e c i m a l p o i n t 2 places to the right: 725 4 So 58 x 12 1/2 = 725. Multiplying a 2-digit number by 15 1 Select a 2-digit number. 2 Add a zero to it. 3 Divide it by 2. 4 Add the number obtained by dividing by 2 to the last number. Example: 1 The 2-digit number chosen to multiply by 15 is 62. 2 Add a zero: 620. 3 Find half (divide by 2): 620/2 = 310. 4 add: 620 + 310 = 930. 5 So 62 x 15 = 930. See the pattern? 1 If the number to be multiplied by 15 is 36: 2 Add a zero: 360. 3 Find half (divide by 2): 360/2 = 180. 4 add: 360 + 180 = 540. 5 So 36 x 15 = 540. Multiplying a 2-digit number by 18 1 Select a 2-digit number. S 2 Multiply it by 2. e 3 Add one zero. e 4 Subtract the number obtained by multiplying by 2 from the last number. t h Example: e p 1 The 2-digit number chosen to multiply by 18 is 28. a 2 28 x 2 = 56 (multiply by 2). t 3 Add one zero: 560. t 4 Subtract: 560 - 56 = 504. e 5 So
- 93. 18 x 28= 504. r n? 1 If the number to be multiplied by 18 is 46: 2 2 x 46 = 92 (multiply by 2). (Think (40 x 2) + (6 x 2) = 80 + 12 = 92) 3 Add one zero: 920. 4 Subtract: 920 - 92 = 828 (Subtract in easy increments: 920 - 100 = 820, 820 + 8 = 828) 5 So 46 x 18 = 828. Practice this procedure, especially selecting the easiest ways to subtract. Multiplying a 2-digit number by 22 1 Select a 2-digit number. 2 Multiply it by 2. 3 The last digit will be the same. _ _ _ X 4 Add the digits right to left. Example: 1 The 2-digit number chosen to multiply by 22 is 78. 2 78 x 2 = 156 (multiply by 2). 3 Last digit is 6. _ _ _ 6 4 Add digits right to left. 6 + 5 = 11 _ _ 1 _ 5 + 1 + 1 (carry) = 7 _ 7 _ _ 5 The first digit is the same. 1 _ _ _ 6 So 22 x 7 8 = 1716. See the pattern? 1 The 2-digit number chosen to multiply by 22 is 34. 2 34 x 2 = 68 (multiply by 2). 3 Last digit is 8. _ _ 8 4 Add digits right to left. 8 + 6 = 14 _ 4 _ 5 First digit + carry: 6 + 1 = 7 7 _ _ 6. So 22 x 34 = 748. Practice getting products by adding digits right to left. Then you will be able to multiply by 22 in your head with ease. Multiplying a 2- or 3-digit number by 25 1 Select a 2- or 3-digit number. (Choose larger numbers when you feel sure about the method.) 2 Divide by 4 (or by 2 twice).
- 94. 3 Add 2zeros (or move the decimal point 2 places to the right). Example: 1 The 2-digit number chosen to multiply by 25 is 78. 2 Divide by 2 twice: 78/2 = 39, 39/2 = 19.5 3 Move the decimal point 2 places to the right: 1950 4 So 78 x 25 = 1950. See the pattern? 1 The 3-digit number chosen to multiply by 25 is 258. 2 Divide by 2 twice: 258/2 = 129, 129/2 = 64.5 3 Move the decimal point 2 places to the right: 6450 4 So 258 x 25 = 6450 Multiplying a 2-digit number by 27 1 1 Select a 2-digit number . 8 2 Multiply it by 3. 9 3 Add one zero. 0 4 Subtract the number obtained by multiplying by 3 from the last number. -1 Example: 8 9 1 The 2-digit number chosen to multiply by 27 is 42. = 2 3 x 42 = 120 + 6 = 126 (multiply by 3). 1 3 Subtract left to right in steps: 1260 - 100 - 20 – 6 = 1160 - 20 - 6 = 1140 - 6 = 1134 or 8 1260 - 130 + 4 - 1130 + 4 = 1134. 9 4 So 42 x 27 = 1134. 0 -See the pattern? 1 0 1 If the number to be multiplied by 27 is 63: 0 2 3 x 63 = 189 (multiply by 3). -3 Add one zero: 1890. 8 4 Subtract: 1890 – 1 89 = 1701 0 (Subtract left to right in steps: -9 = 1790 - 80 - 9 = 1710 - 9 = 1701 or 1890 - 190 + 1 = 1700 + 1 = 1701. 5. So 63 x 27 = 1701. Multiplying a 2-digit number by 33 1 Select a 2-digit number. 2 Multiply it by 3. 3 Add the digits from right to left (see examples, below). Example: 1 The 2-digit number chosen to multiply by 33 is 38. 2 Multiply by 3: 3 x 38 = 90 + 24 = 114 3 The last digit is 4: _ _ _ 4. 4 Add right to left. 4 + 1 = 5: _ _ 5 _ 1 + 1 = 2: _ 2 _ _ First digit is 1: 1 _ _ _ 5 So 33 x 38 = 1254. See the pattern?
- 95. 1 If thenumber to be multiplied by 33 is 82: 2 Multiply by 3: 3 x 82 = 246 3 The last digit is 6: _ _ _ 6. 4. Add right to left. 6 + 4 = 10: _ _ 0 _ 4 + 2 + 1 (carry) = 7: _7__ First digit is 2: 2 _ _ _ 4 So 33 x 82 = 2706. Challenge a friend and BEAT THE CALCULATOR! le: 1 The 2-digit number chosen to multiply by 33 1/3 is 48. 2 Add two zeros: 4800 3 Divide by 3: 4800/3 = 1600 4 So 48 x 33 1/3 = 1600. See the pattern? 1 The 2-digit number chosen to multiply by 33 1/3 is 92. 2 Add two zeros: 9200 3 Divide by 3: 9200/3 = 3066 2/3. 4 So 92 x 33 1/3 = 3066 2/3. Multiplying a 2-digit number by 35 Select a 2-digit number. 1 Multiply it by 7. 2 Divide by 2. 3 Add a zero. Multiplying a 2-digit number by 33 1/3 1 2 Select a 2-digit number . Add two zeros. 3 Divide by 3. Example: 1 The 2-digit number chosen to multiply by 35 is 28. 2 7 x 28 = 140 + 56 = 196. 3 Divide by 2: 196/2 = 98. 4 Add one zero: 980. 5 So 35 x 28 = 980. See the pattern? 1 The 2-digit number chosen to multiply by 35 is 68. 2 7 x 68 = 420 + 56 = 476. 3 Divide by 2: 476/2 = 238. 4 Add one zero: 2380. 5 So 35 x 68 = 2380. Practice and you will be able to give these products easily and accurately.
- 96. Multiplying a 2-digitnumber by 36 1S 1. Select a 2-digit number. 2 Multiply it by 4. 3 Add one zero . 4 Subtract the number obtained by multiplying by 4 from the l ast number. Example: 1 The 2-digit number chosen to multiply by 36 is 52. 2 4 x 52 = 208 (multiply by 4). 3 Add one zero: 2080. 4 2080 - 208 = 1872 ( Subtract left to right in steps: 2080 - 200 - 8 = 1880 - 8 = 1872) 5 So 52 x 36 = 1872. See the pattern? 1 If the number to be multiplied by 36 is 86: 2 4 x 86 = 344 (multiply by 4): (Think 4 x 86 = (4 x 80) + (4 x 6) = 320 + 24 = 344.) 3 Add one zero: 3440. 4 3440 - 344 = 3096 (Subtract left to right in steps: 3440 - 300 - 40 - 4= 3140 - 40 - 4 = 3100 - 4 = 3096) 5. So 86 x 36 = 3096. Mastering the subtraction from left to right will enable you to produce these products of 36 and 2-digit numbers quickly and accurately. Multiplying a 2-digit number by 37 1/2 1T h e 2 1 Select a 2-digit number. -2 Multiply it by 3. d 3 Divide by 8. i 4 Move the decimal point two places to the right. g it Example: n umber chosen to multiply by 37 1/2 is 68. 2 Multiply by 3. 3 x 68 = 180 + 24 = 204 3 Divide by 8. 204/8 = 25.5 4 Move the decimal point two places to the right: 2550 5 So 37 1/2 x 68 = 2550. See the pattern?
- 97. 1 The 2-digitnumber chosen to multiply by 37 1/2 is 28. 2 Multiply by 3. 3 x 28 = 60 + 24 = 84 3 Divide by 8. 84/8 = 10.5 4 Move the decimal point two places to the right: 1050 5 So 37 1/2 x 28 = 1050. Multiply from left to right for easy mental products. Multiplying a 2-digit number by 44 1 Select a 2-digit number. 2 Multiply it by 4. 3 Add the digits from right to left. Example: 1 The 2-digit number chosen to multiply by 44 is 36. 2 4 x 36 = 120 + 24 = 144. 3 The last digit is 4: _ _ _ 4 4 Add the digits in 144 right to left: 4 + 4 = 8 _ _ 8 _ 4 + 1 = 5: _ 5 _ _ First digit is 1: 1 _ _ _ 5 So 44 x 36 = 1584. See the pattern? 1 The 2-digit number chosen to multiply by 44 is 69. 2 4 x 69 = 240 + 36 = 276. 3 The last digit is 6: _ _ _ 6 4 Add the digits in 276 right o left: 6 + 7 = 13 _ _ 3 _ 7 + 2 + 1 (carry) = 10: _ 0 _ _ First digit + 1 (carry) = 2 + 1 = 3: 3___ 5. So 44 x 69 = 3036. Multiplying a 2-digit number by 45 1 Select a 2-digit number. 2 Multiply it by 5. 3 Add a zero. 4 Find the difference between these two numbers Example: 1 The 2-digit number chosen to multiply by 45 is 54. 2 5 x 54 = 270.
- 98. 3 Add onezero: 2700. 4 Subtract: 2700 - 270 = 2430 (subtract left to right: 2700 - 200 - 709 = 2500 - 70 = 2430) 5 So 54 x 45 = 2430. See the pattern? 1 If the number to be multiplied by 45 is 27: 2 5 x 27 = 135 Think 5 x 27 = (5 x 20) + (5 x 7) = 100 + 35 = 135. 3 Add one zero: 1350. 4 Subtract: 1350 - 135 = 1215 (subtract left to right: 1350 - 100 - 30 - 5 = 1250 - 30 - 5 = 1220 - 5 = 1215) 5. So 27 x 45 = 1215. Practice - especially the subtraction - and you will become adept at f inding these products. Multiplying a 2-digit number by 55 (method 1) 31 1 Select a 2-digit number. 4 2 Multiply it by 50. (An easy way to do this is to take half of it (divide by 2), then add two 0 zeros.) 0 3 Add 1/10 (one-tenth) of the product. (Take off the last zero of the product and add this + number to the product.) 1 4 Example: 0 = 1 The 2- digit number chosen to multiply by 55 is 28. 1 2 50 x 28 = 1400 (half of 28 is 14; add 2 zeros). 5 git number chosen to multiply by 63 is 28. 2 7 x 28 = 196 3 Add one zero : 1960 4 Subtract: 1960 - 196 = 1764 (subtract left to right: 1960 - 100 – 90 - 6 = 1860 - 90 - 6 = 1770 - 6 = 1764) 5 So 28 x 63 = 1764. See the pattern? 1 The 2-digit number chosen to multiply by 63 is 83. 2 7 x 83 = 581 (think
- 99. (7x80) + (7x3)= 560 + 21 = 581) 3 Add one zero: 5810 4 Subtract: 5810 – 581 = 5229 (subtract left to right: 5810 - 500 - 80 - 1 = 5310 - 80 – 1 = 5230 - 1 = 5229) 5 So 83 x 63 = 5229. Multiplying a 2-digit number by 66 1 Select a 2-digit number. 2 Multiply it by 6. 3 Add the digits from right to left (see examples, below). Example: 1 The 2-digit number chosen to multiply by 66 is 54. 2 Multiply by 6: 6 x 54 = 300 + 24 = 324 3 The last digit is 4: _ _ _ 4 4 Add right t o left. 4 + 2 = 6: _ _ 6 _ 2 + 3 = 5: _ 5 _ _ First digit is 3: 3 _ _ _ 5 So 66 x 54 = 3564. See the pattern? 1 If the number to be multiplied by 66 is 92: 2 Multiply by 6: 6 x 92 = 540 + 12 = 552 3 The last digit is 2: _ _ _ 2 4. Add right to left. 2 + 5 = 7: _ _ 7 _ 5 + 5 = 10 (carry 1) : _ 0 _ _ First digit + carry: 5 + 1 = 6: 6 _ _ _ 5 So 66 x 92 = 6072. Challenge a friend and BEAT THE CALCULATOR! Multiplying a 2-digit number by 72 m 1 Select a 2-digit number. u 2 Multiply it by 8. lt 3 Add one zero. i 4 Subtract the first number from the second. p li Example: c a 1 Th e 2-digit number chosen to multiply by 72 is 54. ti 2 Multiply by 8: 5 4 x 8 = 432 o (Think: (8 x 50) + (8 x 4) = 400 + 32 = 432) n b 3 Add one zero: 4320 y 4 Subtract: 4320 - 432 = 3888 (Subtract left
- 100. to right: 4320- 400 - 30 - 2 = 3920 - 30 - 2 = 7 3890 - 2 = 3888) 2 5 So 54 x 72 = 3888. See the pattern? 1 The 2-digit number chosen to multiply by 72 is 21. 2 Multiply by 8 : 21 x 8 = 168 3 Add one zero: 1680 4 Subtract: 1680 - 168 = 1512 (Subtract left to right: 1680 - 100 - 60 - 8 = 1580 - 60 - 8 = 1520 – 8 = 1512) 5 So 21 x 72 = 1512. Practice, practice, practice and y ou will become perfect in your Multiplying a 2- or 3-digit number by 75 1 Select a 2- or 3-digit number. 2 Multiply it by 3. Th 3 Add two zeros. e 4 Divide by 4. 3 dig Example: it nu The 2-digit number chosen to multiply by 75 is 62. mb Multiply by 3: 3 x 62 = 186 er Add two zeros: 18600 ch Divide by 4 (or by 2 twice): 18600/2 = 9300, 9300/2 = 4650 os So 75 x 62 = 4650. en to See the pattern? mu The 3-digits number ltiply by 75 is 136. 1 Multiply by 3: 3 x 136 = 390 + 18 = 408 2 Add two zeros: 40800 3 Divide by 4 (or by 2 twice): 40800/2 = 20400, 20400/2 = 10200 4 So 75 x 136 = 10200. Practice and you will be able to give these products quickly. Multiply from left to right for simpler mental math, and divide by 2 twice if division by 4 is not easy Multiplying a 2-digit number by 77 1 Select a 2-digit number. 2 Multiply it by 7. 3 Add the digits from right to left (see examples, below).
- 101. Example: 1 The 2-digitnumber chosen to multiply by 77 is 42. 2 Multiply by 7: 7 x 42 = 280 + 14 = 294 3 The last digit is the same: _ _ _ 4 4 Add right to left. 4 + 9 = 13: _ _ 3 _ 9 + 2 + 1(carry) = 12: _ 2 __ First digit + carry: 2 + 1 = 3: 3 _ _ _ 5 So 77 x 42 = 3234. See the pattern? 1 If the number to be multiplied by 77 is 84: 2 Multiply by 7: 7 x 84 = 560 + 28 = 588 3 The last digit is the same: _ _ _ 8 4 Add right to left. 8 + 8 = 16: _ _ 6 _ 8 + 5 + 1(carry) = 14: _ 4 _ _ first digit + carry: 5 + 1 = 6: 6 _ _ _ 5. So 77 x 84 = 6468. Multiplying a 2-digit number by 81 1 Select a 2-digit number. 2 Multiply it by 9. 3 Add one zero. 4 Subtract the first number from the second. Example: 1 The 2-digit number chosen to multiply by 81 is 31. 2 Multiply by 9: 9 x 31 = 279 3 Add one zero: 2790 4 Subtract: 2790 - 279 = 251 1 (Think: 2790 - 200 - 70 - 9 = 2590 - 70 - 9 = 2520 - 9 = 2511) 5 So 31 x 81 = 2511. See the pattern? 1 The 2-digit number chosen to multiply by 81 is 68. 2 Multiply b y 9: 9 x 68 = 612 (Think: 9 x 6 + 9 x 8 = 540 + 72 = 612) 3 Add one zero: 6120 4 Subtract: 6120 - 612 = 5508 (Think: 6120
- 102. 600 - 10- 2 = 5520 - 10 - 2 = 5510 - 2 = 5508) 5. So 68 x 81 = 5508. Practice subtracting mentally by doing it in steps from left to right . With that skill you will be able to multiply by 81 quickly. Multiplying a 2-digit number by 88 1 1 Select a 2-digit number. 2 Multiply it by 8. T 3 Add the digit from right to left. h e Example: 2 - 1 the 2 digit number chosen to multiply by 88 is 43. 2 Multiply by 8: 8 x 43 = 320 + 24 = 344. 3 The last digit is the same: _ _ _ 4 4 Add the digits in 344 right to left: 4 + 4 = 8 _ _ 8 _ 3 + 4 = 7: _ 7 _ _ First digit is the same: 3 _ _ _ 5 So 88 x 43 = 3874. See the pattern? 1 The 2-digit number chosen to multiply by 88 is 82. 42 Multiply by 8: 8 x 82 = 640 + 16 = 656. 3 The last digit is the same: _ _ _ 6 5 Add the digits in 656 right o left: 6 + 5 = 11 _ _ 1 _ 5 + 6 + 1 (carry) = 12: _ 2 _ _ First digit + 1 (carry) = 6 + 1 = 7: 7 _ _ _ So 88 x 82 = 7216. Multiplying a 2-digit number by 99 1 Select a 2-digit number. 2 Add two zeros to the number. 3 Subtract the original number from the second number. Example: 1 The 2-digit number chosen to multiply by 99 is 79. 2 Add two zeros: 7900. 3 Subtract the 2-digit number: 7900 - 79 = 7900 – 80 + 1 = 7820 + 1 = 7821. 4 So 79 x 99 = 7821.
- 103. See the pattern? hosento multiply by 99 is 42. 2 Add two zeros: 4200. 3 Subtrac the 2-digit number: 4200 - 42 = 4200 - 40 - 2 = 4160 - 2 = 4158. 4 So 42 x 99 = 4158. Multiplying a 2-digit number by 101 1 Select a 2-digit number . 2 Write it twice! Examples: 1 47 x 101 = 4747. 2 38 x 101 = 3838. 3 96 x 101 = 9696. Multiplying a 3-digit number by 101 1 Select a 3-digit number . 2 The sum of the first and third digits will be the middle digit: _ _ X _ _. 3 The first two digits plus the carry will be the first digits: X X _ _ _. 4 The last two digits of the Example: 1 The number chosen is 318. 2 3 + 8 = 11 (sum of first and third digits): _ _ 1 _ _ (keep carry, 1) 3 31 + 1 = 32 (first two digits plus carry): 3 2 _ _ _. 4 The last two digits are the same: _ _ _ 1 8. 5 So 318 x 101 = 32118. See the pattern? 1 If the number chosen is 728: 2 7 + 8 = 15 (sum of first and third digits): _ _ 5 _ _ (keep carry, 1) 3 72 + 1 = 73 (first two digits plu s carry): 7 3 _ _ _. 4T h e l a s t two digits are the same: _ _ _ 2 8. 5 So 728 x 101 = 73528. Multiplying a 2-digit number by 1 1/2 1 Select a 2-digit number. 2 Multiply by 3. 3 Divide by 2. Example:
- 104. 1 The 2-digitnumber chosen to multiply by 1 1/2 is 63. 2 Multiply by 3: 3 x 63 = 180 + 9 = 189 3 Divide by 2: 189/2 = 94 1/2 4 So 63 x 1 1/2 = 94 1/2. See the pattern? 1 If the 2-digit number chosen to multiply by 1 1/2 is 97: 2 Multiply by 3: 3 x 97 = 270 + 21 = 291 3 Divide by 2: 291/2 = 145 1/2 4 So 9 7 x 1 1/2 = 145 1/2. Remember, if you use a calculator you will get a decimal approxi mation, whereas the fractional form will be exact. 1 Multiplying a 2-digit number by 1 1/3 Select a 2-digit number. Multiply by 4. Divide by 3. Example: 5 The 2-digit number chosen to multiply by 1 1/3 is 32. 6M u lt i p l y b y 4: 4 x 32 = 120 + 8 = 128 7 Divide by 3: 128/3 = 42 2/3 8 So 32 x 1 1/3 = 42 2/3. See the pattern? If the 2-digit number chosen to multiply by 1 1/3 is 83: Multiply by 4: 4 x 83 = 320 + 12 = 332 Divide by 3: 332/3 = 110 2/3 So 83 x 1 1/ 3 = 110 2/3. Remember, if you use a calculator you will get a decimal approxi mation, whereas the fractional form will be exact. Multiplying a 2-digit number by 1 1/5
- 105. 5 Select a2-digit number. 6 Multiply the number by 6. 7 Divide the p roduct by 5. Example: 8 The 2-digit number chosen to multiply by 1 1/5 is 34. 9 Multiply by 6: 6 x 34 = 180 + 24 = 204 10 Divide the product by 5: 204/5 = 40.8 11 S o 34 x 1 1/5 = 40.8. See the pattern? 12 The 2-digit number chosen to multiply by 1 1/5 is 61. 13 Multiply b y 6: 6 x 61 = 366 14 Divide the product by 5: 366/5 = 73.2 15 So 34 x 16 Select a 2-digit number. 17 Divide by 6 (or by 2 and 3). 18 M o v e t h e decimal point one place to the right, or add a zero. Example: 19 The 2-digit number chosen to multiply by 1 2/3 is 78. 20 Divide by 2: 78/2 = 39 21 Divide by 3: 39/3 = 13. (39 is divisible by 3 because the sum of its digits is divisible by 3.) 22 Add a zero: 130. 23 So 78 x 1 2/3 = 130. See the pattern? 24 If the 3-digit number chosen to multiply by 1 2/3 is 315: 25 Divide by 3: 315 / 3 = 105. (315 is divisible by 3 because the sum of its digits is divisible by 3.) 26 Divide by 2: 105/2 = 52.5. 27 Move the decimal point one place to the right: 525.28 So 315 x 1 2/3 = 525. Test the number to see if it is divisible by 3 (if the sum of the digits is divisible by
- 106. Multiplying a 2-digitnumber by 125 29 Select a 2-digit number. 30 Divide the number by 8. 31 Move the decimal point 3 places to the right (add three zeros). Example: 32 The 2-digit number chosen to multiply by 125 is 34. 33 Divide by 8: 34/8 = 4.25 34 Move the decimal point 3 places to the right: 4250 35 So 125 x 34 = 4,250. See the pattern? 36 The 2-digit number chosen to multiply by 125 is 78. 37 Divide by 8: Multiplying a 2-digit number by 198 5 Select a 2-digit number. 6 Double the number. 7 If the double has 2 digits, subtract 1: X X _ _ Subtract the double from 100: _ _ XX 8 If the double has 3 digits, subtract 2: X X X _ _ (if it has 2 zeros, su btract the first digit) Subtract the last 2 digits of the double from 100: _ _ _ X X Example: 9 The 2-digit number chosen to multiply by 198 is 21. 10 Double it: 2 x 21 = 42. 11 Subtract 1: 42 - 1 = 41: 4 1 _ _ 12 100 - double: 100 - 42 = 58: _ _ 5 8 13 So 198 x 21 = 4158. See the pattern? 1. The 2-digit number chosen to multiply by 198 is 56. ouble it: 2 x 56 = 112. Subtract 2: 112 - 2 = 110: 1 1 0 _ _ 100 - double's last 2 digits: 100 - 12 = 88: _ _ _ 8 8 So 198 x 56 = 11088. Multiplying a 2- or 3-digit number by 2 1/2
- 107. 14 Select a2- or 3-digit number. S 15 Divide by 4 (or by 2 twice). e 16 Move the decimal point one place to the right, or add a zero. e t Example: h e 17 If the 2-digit number chosen to multiply by 2 1/2 is 98 18 Divide by 2: 98/2 = 49 a p 19 Divide by 2: 49/2 = 24.5. t20 Move the decimal point one place to the right: 245 t21 So 98 x 2 1/2 = 245. e rn? 22 If the 3-digit number chosen to multiply by 2 1/2 is 123 23 Divide by 2: 123/2 = 61.5 24 Divide by 2: 61.5/2 = 30.75. 25 Move the decimal point one place to the right: 307.5 26 So 123 x 2 1/2 = 307.5. 27 If the 3-digit number chosen to multiply by 2 1/2 is 488 Divide by 4: 488/4 = 122 Divide by 2: 61.5/2 = 30.75. Move the decimal point one place to the right (add a zero): 1220. So 488 x 2 1/2 = 1220. Divide by 4 when it is easy (the number is divisible by 4 if its last two d igits are divisible by 4), or divide by 2 twice. This method should bec me an easy one. Multiplying a 2-digit number by 2 1/3 28 Select a 2-digit number. 29 Multiply the number by 7. 30 Divide by 3. Example: 31 The 2-digit number chosen to multiply by 2 1/3 is 67. 32 Multiply by 7: 7 x 67 = 420 + 49 = 469. 33 Divide by 3: 469/3 = 156 1/3. 34 So 67 x 2 1/3 = 156 1/3.
- 108. See the pattern? 35The 2-digit number chosen to multiply by 2 1/3 is 91. 36 Multiply by 7: 7 x 91 = 630 + 7 = 637. 37 Divide by 3: 637/3 = 212 1/3. 38 So 91 x 2 1/3 = 212 1/3. Multiplying a 2-digit number by 2 1/4 39 Select a 2-digit number. 40 Multiply the number by 9. 41 Divide by 4. Example: 42 The 2-digit number chosen to multiply by 2 1/4 is 51. 43 Multiply by 9: 9 x 51 = 450 + 9 = 459. 44 Divide by 4: 459/4 = 114 3/4. 45 So 51 x 2 1/4 = 114 3/4. See the pattern? 46 The 2-digit number chosen to multiply by 2 1/3 is 28. 47 Multiply by 9: 9 x 28 = 180 + 72 = 252. 48 Divide by 4: 252/4 = 63. 49 So 51 x 2 1/4 = 63. Multiplying a 2-digit number by 2 1/5 50 Select a 2-digit number. 51 Multiply the number by 11. Add the digits r ight to left. 52 Divide the product by 5. Example: 53 The 2-digit number chosen to multiply by 2 1/5 is 43. 2. Multiply by 11: the last digit of the product is 3. 3 + 4 = 7 (next left digit). The product is 473. 54 Divide the product by 5: 473/5 = 94.6. 55 So 43 x 2 1/5 = 94.6. See the pattern? 56 The 2-digit number chosen to multiply by 2 1/5 is 78. 2. Multiply by 11: the last digit of the product is 8. 8 + 7 = 15. The ne
- 109. xt digit is5, carry 1. 7 + 1 = 8. The product is 858. 57 Divide the product by 5: 858/5 = 171.6. 58 So 78 x 2 1/5 = 171.6. Multiplying a 2-digit number by 2 2/3 59 Select a 2-digit number. 60 Multiply the number by 8. 61 Divide by 3. Example: 62 The 2-digit number chosen to multiply by 2 2/3 is 24. 63 Multiply by 8: 8 x 24 = 160 + 32 = 192. 64 Divide by 3: 192/3 = 64. 65 So 24 x 2 2/3 = 64. See the pattern? 66 The 2-digit number chosen to multiply by 2 2/3 is 68. 67 Multiply Multiplying a 2-digit number by 2 2/9 81 A d d o 70 Select a 2-digit number. n 71 Multiply by two. e 72 Add one zero. z 73 Divide by nine. e r Example: o : 74 The 2-digit number chosen to multiply by 2 2/9 is 31. 1 75 Multiply by 2: 2 x 31 = 62 76 Add one zero: 620 77 Divide by 9: 620/9 = 68 8/9 78 So 31 x 2 2/9 = 68 8/9. See the pattern? 4 8 0 82 D i v i d 79 The 2-digit number chosen to multiply by 2 2/9 is 74. 80 Multiply by 2: 2 x 74 = 148 b e y 9: 1480/9 = 164 4/9 83 So 74 x 2 2/9 = 164 4/9. Those using calculators will have difficulty entering 2 2/9 (2.222222...
- 110. ) and their ultsmay be repeating decimals. Your steps will produce exact results. Multiplying a 2-digit number by 250 84 Select a 2-digit number. 85 Divide the number by 4. 86 Move the decimal point 3 places to the right (add three zeros). Example: 87 The 2-digit number chosen to multiply by 250 is 96. 88 Divide by 4: 96/4 = 24 89 Add three zeros: 24000 90 So 96 x 250 = 24,000. See the pattern? 91 The 2-digit number chosen to multiply by 250 is 37. 92 Divide by 4: 37/4 = 9.25 93 Move the decimal point 3 places to the right: 9250 94 S o 37 x 250 = 9,250. Multiplying a 2-digit number by 297 95 Select a 2-digit number. 96 Multiply by 3. 97 If the product has 2 digits, subtract 1: X X _ _ Subtract the product from 100: _ _ XX 98 If the product has 3 digits, add 1 to the first digit: X X X _ _ Subtract this from the product; Subtract the last 2 digits of the product from 100: ___XX Example: 99 T he 2-digit number chosen to multiply by 297 is 33. 100 Multiply by 3: 3 x 33 = 99. 101 Subtract 1: 99 - 1 = 98: 9 8 _ _ 102 100 - product: 100 - 99 = 1: _ _ 0 1 103 So 297 x 33 = 9801. See the pattern?
- 111. 1. The 2-digitnumber chosen to multiply by 297 is 92. 104 Multiply by 3: 3 x 92 = 276. 105 Add 1 to the first digit: 2 + 1=3 106 Subtract this from the product: 276-3 = 273: 2 7 3 _ _ 107 Subtract the last two digits of the product from 100: 100 - 76 = 24: _ _ _ 2 4 108 So 297 x 92 = 27324.. Multiplying a 2-digit number by 3 1/2 109 Select a 2-digit number. 110 If number is even, divide by 2, then multiply by 7. 111 If number is odd, multiply by 7, then divide by 2. Example: 4 The 2-digit number chosen to multiply by 3 1/2 is 84. 5 Divide by 2: 84/2 = 42 6 Multiply by 7: 42 x 7 = 280 +14 = 294 7 So 84 x 3 1/2 = 294. See the pattern? 112 The 2-digit number chosen to multiply by 3 1/2 is 63. 113 Multiply by 7: 63 x 7 = 420 + 21 = 441 114 Divide by 2: 441/2 = 220.5 115 So 63 x 3 1/2 = 220.5. Multiplying a 2-digit number by 3 1/3 118 M 116 Select a 2-digit number. o 117 Divide the number by 3. v e the decimal point one place to the right (or add a zero). Example: 119 The 2-digit number chosen to multiply by 3 1/3 is 78. 120 Divide by 3: 78/3 = 26 121 Add a zero: 260 122 So 78 x 3 1/3 = 260. See the pattern? 123 The 2-digit number chosen to multiply by 3 1/3 is 29. 124 Divide by
- 112. 3: 29/3 =9.6 2/3 125 Move the decimal point one place to the right: 96 2/3 126 So 29 x 3 1/3 = 96 2/3. If the division does not come out even, divide to one decimal place and express the result as a fraction. Your answer will be a mixed number when you move the decimal point one place to the right. Multiplying a 2-digit number by 3 3/4 127 Select a 2-digit number. 128 Multiply the number by 30. 129 Divide by 8. Example: 130 The 2-digit number chosen to multiply by 3 3/4 is 24. 131 Multiply by 30: 24x30 = 600+120 = 720. 132 Divide by 8: 720/8 = 90. 133 So 24 x 3 3/4 = 90. See the pattern? 134 The 2-digit number chosen to multiply by 3 3/4 is 52. 135 Multiply by 30: 52x30 = 1500+60 = 1560. 136 Divide by 8: 1560/8 = 195. 137 So 52 x 3 3/4 = 195. number by 375 138 Select a 2-digit number. 139 Multiply by 3. 140 Divide by 8. 141 Example: 142 The 2-digit number chosen to multiply by 375 is 38. 143 Multiply by 3: 3 x 38 = 90 + 24 = 114 144 Divide by 8: 114/8 = 14 2/8 14.25 145 Move the decimal point 3 places to the right: 14250 146 So 375 x 38 = 14250. See the pattern? 147 The 2-digit number chosen to multiply by 375 is 64. 148 Multiply
- 113. by 3: 3x 64 = 180 + 12 = 192 149 Divide by 8: 192/8 = 24 150 Move the decimal point 3 places to the right: 24000 151 So 375 x 64 = 24000. Multiply left to right to get products, and express the remainder as a fraction that is easily written as a decimal. Multiplying a 2-digit number by 396 S 152 Select a 2-digit number. u 153 Multiply by 4. b 154 If the product has 2 digits, subtract 1: X X _ _ tr Subtract the product from 100: _ _ X X a c 155 If the product has 3 digits, add 1 to the first digit: t X X X _ _ t Subtract this from t he product; h e last 2 digits of the product from 100: _ _ _ X X 156 If the product has two final zeros, subtract the 1st digit from the product: X X X _ _ The last 2 digits are 0 0 : _ _ _ 0 0 Example: 157 The 2-digit number chosen to multiply by 396 is 18. 158 Multiply by 4: 4 x 18 = 40 + 32 = 72 159 Subtract 1: 72 - 1 = 71: 7 1 _ _ 160 100 - product: 100 - 72 = 28: _ _ 2 8 161 So 18 x 396 = 7128. See the pattern? 162 The 2-digit number chosen to multiply by 396 is 68. Multiply by 4: 4 x 68 = 240 + 32 = 272 1st digit + 1: 2 + 1 =3 Subtract from product: 272 - 3 = 269: 2 6 9 _ _ 100 - product's last 2 digits: 100 - 72 = 28: _ _ 2 8 So 69 x 396 = 26928.
- 114. 163 The 2-digitnumber chosen to multiply by 396 is 50. 164 Multiply by 4: 4 x 50 = 200 165 Subtract 1st digit: 200 - 2 = 198: 1 9 8 _ _ 166 Last 2 digits are 0 0: _ _ _ 0 0 167 So 50 x 396 = 19800. Practice multiplying by 4 from left to right. Repeat the first digits of the answer, then add the final two. Your speed with this large multiplication problem will be impressive. Multiplying a 2-digit number by 4 1/2 168 Select a 2-digit number. 169 Multiply by 9. 170 Divide by 2. Example: 171 The 2-digit number chosen to multiply by 4 1/2 is 52. 172 Multiply by 9: 52 x 9 = 450 +18 = 468 173 Divide by 2: 468/2 = 234 174 S o 5 2 x 4 1 / 2 = 234. See the pattern? 175 The 2-digit number chosen to multiply by 4 1/2 is 91. 176 Multiply by 9: 91 x 9 = 810 + 9 = 819 177 Divide by 2: 819/2 = 409 1/2 178 So 91 x 4 1/2 = 409 1/2. Multiplying a 2-digit number by 495 179 Select a 2-digit number . 180 Multiply by 5. 181 If the product ha s 2 digits, subtract 1: X X _ _ Subtract the product from 100: __XX 182 If the product has 3 digits, add 1 to the first digit: X X X _ _ (don't add 1 if the last two digits are zeros) Subtract this from the product: X X X _ _ Subtract the last 2 digits of the product from 100: _ __XX
- 115. Example: 183 The 2-digitnumber chosen to multiply by 495 is 18. 184 Multiply by 5: 5 x 18 = 90. 185 Subtract 1: 90 - 1 = 89: 8 9 _ _ 186 100 – produ ct: 100 - 90 = 10: _ _ 1 0 187 So 495 x 18 = 8910. See the pattern? 188 The 2-digit number chosen to multiply by 495 is 67. 190 189 Multiply by 5: 5 x 67 = 300 + 35 = 335. 191 Add 1 to the first digit :3+1=4 Subtract this from the product: 335-4 = 331: 3 3 1 _ _ Subtract the last two digits of the product from 100: 100 - 35 = 65: _ _ _ 6 5 So 495 x 67 = 33165. Multiplying a 2-digit number by 5 5/7 192 Multiply the number by 4. 193 Add a zero. 194 Divide by 7. Example: 195 If the 2-digit number chosen to multiply by 5 5/7 is 32: 196 M ultiply by 4: 4 x 32 = 128 197 Add a zero: 1280 198 Divide by 7: 1280/ 7 = 182 6/7 199 So 32 x 5 5/7 = 182 6/7. See the pattern? 200 If the number to be multiplied by 5 5/7 is 61: 201 Multiply by 4: 4 x 61 = 244 202 Add a zero: 2440 203 Divide by 7: 2440/7 = 348 4/7 204 So 61 x 5 5/7 = 348 4/7. Numbers divisible by 7 will produce whole-number answers. With
- 116. other numbers, thoseusing calculators will get long, inexact decimals. Multiplying a 2-digit number by 5 5/9 214 M 205 Multiply the number by 5. u 206 Add a zero. lt 207 Divide by 9. i pExample: l y 208 If the 2-digit number chosen to multiply by 5 5/9 is 36 : b 209 Multiply by 5: 5 x 36 = 150 + 30 = 180 y 210 Add a zero: 1800 5 211 Divide by 9: 1800/9 = 200 : 212 So 5 5/7 x 36 = 200. 5 x See the pattern? 7 1 213 If the 2-digit number chosen to multiply by 5 5/9 is 71: = 355 215 Add a zero: 3550 216 Divide by 9: 3550/9 = 394 4/9 217 So 5 5/9 x 71 = 394 4/9. Numbers divisible by 9 will produce whole-number answers. With other numbers, those using calculators will ge Multiplying a 2-digit number by 594 232 S o 5 9 218 Select a 2-digit number . 4 219 Multiply it by 6. x 220 If the product is 2 digits, subtract 1: 9 X X _ _ 2 Then subtract the product from 100: = __XX5 4 221 If the product is 3 digits, add one to the first digit and subtract the r esult from the product: , X X X _ _ Then subtract the last two digits of the product from 100: _ _ _ X X 6 4 Example: 8 . 222 The 2-digit number chosen to multiply by 594 is 16. 223 Multiply by 6:
- 117. 6 x 16= 60 + 36 = 96 224 Subtract 1: 96 - 1 = 95: 9 5 _ _ 225 100 - product: 100 - 96 = 4: _ _ 0 4 226 So 594 x 16 = 9,504. See the pattern? 227 The 2-digit number chosen to multiply by 594 is 92. 228 Multiply by 6 : 6 x 92 = 540 + 12 = 552 229 1st digit + 1: 5 + 1 = 6 230 Subtract 6 from product: 552 - 6 = 546: 5 4 6 _ _ 231 100 - product's last 2 digits: 100 - 52 = 48: _ _ _ 4 8 vide by 4. 235 Divide the resulting quotient by 4. 236 Move the decimal point two places to the right. Example: 237 The 2-digit number chosen to multiply by 6 1/4 is 56. 238 Divide by 4: 56/4 = 14 239 Divide by 4 again: 14/4 = 3.5 240 Move the decimal point two places to the right: 350 241 So 56 x 6 1/4 = 350. See the pattern? 242 The 2-digit number chosen to multiply by 6 1/4 is 82. 243 Divide by 4: 82/4 = 20.5 244 Divide by 4 again: 20.5/4 = 5.125 245 Move the decimal point two places to the right: 512.5 246 So 82 x 6 1/4 = 512.5. ultiplying a 2-digit number by 6 2/3 247 Select a 2-digit number. 248 Multiply the number by 2. 249 Add a zero. 250 Divide by 3. Example: 251 The 2-digit number chosen to multiply by 6 2/3 is 63. 252 Multiply by
- 118. 2: 2 x63 = 126. 253 Add a zero: 1260. 254 Divide by 3: 1260/3 = 420. 255 So 63 x 6 2/3 = 420. See the pattern? 256 The 2-digit number chosen to multiply by 6 2/3 is 41. 257 Multiply by 2: 2 x 41 = 82. 258 Add a zero: 820. 259 Divide by 3: 820/3 = 273 1/3. 260 So 41 x 6 2/3 = 273 1/3. visible by three, the answer will be a whole number. If not, a calculator will get only an approximation, but your answer will be correct. Multiplying a 2-digit number by 625 261 Select a 2-digit number. 262 Add a zero and divide by 2. 263 Divide by 8. 264 Move the decimal point 3 places to the right. Example: 265 The 2-digit number chosen to multiply by 625 is 38. 266 Add zero: 380 ; divide by 2: 380/2 = 190 267 Divide by 8: 190/8 = 23 6/8 = 23.75 268 Move the decimal point 3 places to the right: 23750 269 So 625 x 38 = 23,750. See the pattern? 270 The 2-digit number chosen to multiply by 693 is 72. 271 Add zero: 720; divide by 2: 720/2 = 360 272 Divide by 8: 360/8 = 45 273 Move the decimal point 3 places to the right: 45000 274 So 625 x 72 = 45,000. Multiplying a 2-digit number by 693 279 T 275 Select a 2-digit number . h 276 Multiply it by 7. e 277 If the product is 2 digits, subtract 1: 2 X X _ _ Then subtract the product
- 119. from 100: __ X X d i 278 If the product is 3 digits, add one to the first digit and subtract the result from the product: gX X X _ _ Then subtract the last two digits of the product from 100: _ _ _ X X it n Example: u mber chosen to multiply by 693 is 14. 280 Multiply by 7: 7 x 14 = 70 + 28 = 98 281 Subtract 1: 98 - 1 = 97: 9 7 _ _ 282 100 - product: 100 - 98 = 2: _ _ 0 2 283 So 14 x 693 = 9,702. See the pattern? 284 The 2-digit number chosen to multiply by 693 is 87. 287 285 Multiply by 7: 7 x 87 = 560 + 49 = 609 286 1st digit + 1: 6 + 1 = 7 288 Subtract 7 from product: 609 - 7 = 602: 6 0 2 _ _ 100 - product's last 2 digits: 100 - 9 = 91: _ _ _ 9 1 So 87 x 693 = 60,291. Multiplying a 2-digit number by 792 289 Select a 2-digit number. 290 Multiply it by 8. 291 Add two zeros. 292 Subtract the original number from the result. Example: 293 The 2-digit number chosen to multiply by 792 is 22. 294 Multiply by 8: 8 x 22 = 176 (Think: 8 x 20 + 8 x 22 = 176) 295 Add two zeros: 17600 296 Subtract 17600 - 176 = 17424 (Think: 17600 - 100 - 70 - 6 = 17500 - 70 - 6 = 17430 - 6 = 17424) 297 So 22 x 792 = 17,424. See the pattern? 298 The 2-digit number chosen to multiply by 792 is 91. 301 299
- 120. Multiply by 8:8 x 91 = 728 (Think: 8 x 90 + 8 x 1 = 720 + 8 = 728) 300 Add two zeros: 72800 302 Subtract 72800 - 728 = 72072 (Think: 72800 - 700 - 20 - 8 = 72100 - 20 - 8 = 72080 - 8 = 72072) So 91 x 792 = 72,072. Practice the left-to-right subtraction. Think each step to yourself as a cue for the next. Multiplying a 2-digit number by 8 1/3 303 Select a 2-digit number. 304 Add two zeros. 305 Divide by four. 306 Divide by three. Example: 307 The 2-digit number chosen to multiply by 8 1/3 is 64. 308 Add two zeros: 6400. 309 Divide by four: 6400/4 = 1600. 310 Divide by three: 1600/3 = 555 1/3. 311 So 64 x 8 1/3 = 555 1/3. See the pattern? 312 The 2-digit number chosen to multiply by 8 1/3 is 37. 313 Add two zeros: 3700. 314 Divide by four: 3700/4 = 925. 315 Divide by three: 925/3 = 308 1/3. 316 So 37 x 8 1/3 = 308 1/3. B Multiplying a 2-digit number by 875 317 Select a 2-digit number. 318 Multiply by 7. 319 Divide by 8. 320 Move the decimal point 3 places to the right. Example: 321 The 2-digit number chosen to multiply by 875 is 38. 322 Multiply by 7: 7 x 38 = 210 + 56 = 266 323 Divide by 8: 266/8 = 33 2/8 = 33.25
- 121. 324 Move thedecimal point 3 places to the right: 33250 325 So 87 5 x 38 = 33250. See the pattern? 326 The 2-digit number chosen to multiply by 875 is 74. 327 Multiply by 7: 7 x 74 = 490 + 28 = 518 328 Divide by 8: 518/8 = 64 6/8 = 64.75 329 Move the decimal point 3 places to the right: 64750 330 So 875 x 74 = 64750. Multiplying a 2-digit number by 8 8/9 338 D i v i 331 Select a 2-digit number. d 332 Multiply by eight. e 333 Add one zero. b 334 Divide by nine. y 9 Example: : 3 335 The 2-digit number chosen to multiply by 8 8/9 is 41. 2 336 Multiply by 8: 8 x 41 = 328 8 337 Add one zero: 3280 0 /9 = 364 4/9 339 So 41 x 8 8/9 = 364 4/9. See the pattern? 340 The 2-digit number chosen to multiply by 8 8/9 is 62. 341 Multiply by 8: 8 x 62 = 480 + 16 = 496 342 Add one zero: 4960 343 Divide by 9: 4960/9 = 551 1/9 344 So 62 x 8 8/9 = 551 1/9. Those using calculators will get repeating decimal answers, while your answers will be exact. Multiplying a 2-digit number by 891 345 Select a 2-digit number. 346 Multiply it by 9. 347 If the produc is 2 digits, subtract 1: X X _ _ Then subtract the product from 100: __XX
- 122. 348 If theproduct is 3 digits, add 1 to the first digit and subtract it from the product: X X X _ _ Then subtract the last 2 digits of the product from 100: _ _ _ X X Example: 349 The 2-digit number chosen to multiply by 891 is 11. 350 Multiply by 9: 9 x 11 = 99 351 Subtract 1: 99 - 1 = 98: 9 8 _ _ 352 Subtract the product from 100: 100 - 99 = 1: _ _ 0 1 353 So 11 x 891 = 9,801. See the pattern? 1. The 2-digit number chosen to multiply by 891 is 34. 355 A 354 Multiply by 9: 9 x 34 = 270 + 36 = 306 d d 1 to the first digit: 3 + 1 = 4 356 Subtract 4 from the product: 306 – 4 = 302: 3 0 2 _ _ 357 Subtract the last 2 digits from 100: 100 - 6 = 94: _ _ 9 4 358 So 34 x 891 = 30,294. Multiplying a 2-digit number by 999 359 Select a 2-digit number. 360 Add one zero to the number, subtract 1: X X X _ _ 361 Subtract the original number from 100: _ _ _ X X Example: 362 The 2-digit number chosen to multiply by 999 is 64. 363 Add one zero and subtract 1: 640 - 1 = 639: 6 3 9 _ _ 364 Subtract 64 from 100: 100 - 64 = 36: X X X 3 6 365 So 64 x 999 = 63,936. See the pattern? 8 The 2-digit number chosen to multiply by 999 is 75. 9 Add one zero and subtract 1: 750 - 1 = 749: 7 4 9 _ _ 10 Subtract 75 from 100: 100 - 75 = 25: X X X 2 5 4. So 64 x 999 = 74,925.
- 123. Multiplying a 2-digitnumber by 1001 and adding 21 366 S elect a 2-digit number. 367 Write it twice. 368 Write one zero between the 2-digit numbers. 369 Add 21. Example: 370 12 x 1001 = 12012 12012 + 21 = 12033 371 So 12 x 1001 + 21 = 12033 Multiplying a 3-digit number by 1001 and adding 201 372 Select a 3-digit number. 373 Write it twice. 374 Add 201. Example: 375 123 x 1001 = 123123 376 123123 + 201 = 123324 377 So 123 x 1001 + 201 = 123324 Create variations by changing the number to be added or by giving a number to be subtracted. In either case, make the number an easy one to handle mentally Multiplying a 2- or 3-digit number by 16 2/3 S 378 Select a 2-digit number. (Choose larger numbers when you feel sure about the method.) e 379 Add two zeros to the number. e 380 Divide by 6 (or by 2 and 3). t h Example: e 381 The 2-digit number chosen to multiply by 16 2/3 is 96. a p 382 Add two zeros: 9600. t383 Divide by 3: 9600/3 = 3200. (9600 is divisible by 3 because the sum of its digits is tdivisible by 3.) e 384 Divide by 2: 3200/2 = 1600. r 385 So 96 x 16 2/3 = 1600. n
- 124. ? 386 If the2-digit number chosen to multiply by 16 2/3 is 76: 387 Add two zeros: 7600. 388 Divide by 2: 7600/2 = 3800. 389 Divide by 3: 3800 / 3 = 1266 2/3. 390 So 76 x 16 2/3 = 1266 2/3. Test the number to see if it is divisible by 3. If not, divide by 2 first. With some practice you will become adept at multiplying by 16 2/3 Multiplying a 2-digit number by 10001 and adding 21 The product of a 2-digit number and 10,001 + 21 is: 391 the two digits, 392 two zeros, 393 the two digits, 394 add 21. Example: 29 x 10001 + 21 395 29 x 10001 = 290029 396 290029 + 21 = 290050 Multiplying a 3-digit number by 10001 and adding 201 The product of a 3-digit number and 10,001 + 201 is: 397 the three digits, 398 one zero, 399 the three digits, 4. add 201. Example: 436 x 10001 + 201 437 x 10001 = 4370437 2. 4370437 + 201 = 4370638 Multiplying a 4-digit number by 10001 and adding 2001 The product of a 4-digit number and 10,001 + 2001 is: the four digits, repeated once add 2001. Example: 6741 x 10001 + 2001 6741 x 10001 = 67416741 67416741 + 2001 = 67418742 Create variations by changing the number to be added or subtracted - just make the number an easy one to handle mentally. Multiplying a 2-, 3-, 4-, or 5-digit number f by 100001 i r
- 125. 1. The productof a 2-digit number and 100,001 is: s t first the two digits, then three zeros, t then the two digits again. h e Examples: t h 12 x 100001 = 1200012 r 79 x 100001 = 7900079 e e 2. The product of a 3-digit number and 100,001 is: d i gits, then two zeros, then the three digits again. Examples: 386 x 100001 = 38600386 914 x 100001 = 91400914 3. The product of a 4-digit number and 100,001 is: first the four digits, then one zero, then the four digits again. Examples: 6295 x 100001 = 629506295 8106 x 100001 = 810608106 4. The product of a 5-digit number and 100,001 is: first the five digits, then the five digits again. Examples: 12345X100001 = 1234512345 98765X100001 = 9876598765 Your friends will need a BIG calculator to keep up with you when you announce these products! Multiplying two 2-digit numbers (difference of 2) 400 Choose a 2-digit number. 401 Select a number either 2 smaller or
- 126. 2 larger. 402Square the average of the two numbers. 403 S u b tr a ct 1 from this square. Example: 404 If the first number is 29, choose 31 as the second number. 405 The average of 29 and 31 is 30. Square 30: 30 x 30 = 900. 406 Subtract 1: 900 - 1 = 899. 407 So 29 x 31 = 899. See the pattern? 408 If the first number is 76, choose 74 as the second number. 409 The average of 76 and 74 is 75. Square 75: 75 x 75 = 5625. 410 Subtract 1: 5625 - 1 = 5624. 411 So 76 x 74 = 5624. Practice this shortcut (remember your methods of squaring numbers). Multiplying two 2-digit numbers (difference of 3) 412 Choose a 2-digit number. 413 Select a number either 3 smaller or 3 larger. 414 Add 1 to the smaller number, then square. 415 Subtract one from the smaller number. 416 Add this to the square. Example: 417 If the first number is 27, choose 24 as the second number. 418 Add 1 to the smaller number: 24 + 1 = 25. 419 Square this number: 25 x 25 = 625. 420 Subtract one from the smaller number: 24 - 1 = 23. 421 Add this to the square: 625 + 23 = 648. 422 So 27 x 24 = 648. See the pattern? 423 If the first number is 34, choose 31 as the second number. 427424
- 127. Add 1 tothe smaller number: 31 + 1 = 32. 425 Square this number: 32 x 32 = 1024. 426 Subtract one from the smaller number: 31 - 1 = 30. 428 Add this to the square: 1024 + 30 = 1054. So 34 x 31 = 1054. Choose the second number that will give you the easier square, and use your square shortcuts. Multiplying two 2-digit numbers (difference of 4) 429 Choose a 2-digit number. 430 Select a number either 4 smaller or 4 larger. 431 Find the middle number of the two (the average). 432 Square this middle number. 433 Subtract 4 from this square. Example: 434 If the first number is 63, choose 67 as the second number. 435 The middle number (the average) is 65. 436 Square this middle number: 65 x 65 = 4225. (Remember how to square a 2-digit number ending in 5?) 437 Subtract 4 from this square: 4225 - 4 = 4221. 438 So 63 x 67 = 4221. See the pattern? 439 If the first number is 38, choose 42 as the second number. 440 The middle number (the average) is 40. 441 Square this middle number: 40 x 40 = 1600. 442 Subtract 4 from this square: 1600 - 4 = 1596. 5. So 38 x 42 = 1596. Choose the second number that will give you the easier square, and use
- 128. your square shortcuts. Multiplyingtwo 2-digit numbers 443 C (difference of 6) h o ose a 2-digit number. 444 Select a number either 6 smaller or 6 larger. 445 Find the middle number of the two (the average). 446 Square this middle number. 447 Subtract 9 from this square. Example: 448 If the first number is 78, choose 72 as the second number. 449 The middle number (the average) is 75. 450 Square this middle number : 75 x 75 = 5625. (Remember how to square a 2-digit number ending in 5?) 451 Subtract 9 from this square: 5625 - 9 = 5616. 452 So 78 x 72 = 5616. See the pattern? 453 If the first number is 31, choose 37 as the second number. 456454 The middle number (the average) is 34. 455 Square this middle number : 34 x 34 = 1156. (Remember how to square a 2-digit number ending in 4?) 457 Subtract 9 from this square: 1156 - 9 = 1147. So 31 x 37 = 1147. Choose the second number that will give you the easier square, and use your square shortcuts. Practice! Multiplying two 2-digit numbers 13 S (difference of 8) q u 458 Choose a 2-digit number. a 459 Select a number either 8 smaller or
- 129. 8 larger. r460 Find the middle number of the two (the average). e 461 Square this middle number (multiply it by itself). t 462 Subtract 16 from this square. h i Example: s m 11 If the first number is 34, choose 26 as the second number (8 smaller) . i 12 The middle number (the average) is 30. d dle number: 30 x 30 = 900. 14 Subtract 16 from this square: 900 - 16 = 884 . 15 So 34 x 26 = 884. See the pattern? 463 If the first number is 64, choose 72 as the second number (8 larger). 464 The middle number (the average) is 68. 465 Square this middle number: 68 x 68 = 4624. (Remember how to square a 2-digit number ending in 8?) 466 Subtract 16 from this square: 4624 - 16 = 4608. 5. So 64 x 72 = 4608. Choose the second number that will give you the easier square, and use your square shortcuts. Practice! Multiplying two 2-digit numbers e (difference of 10) n d 467 Choose a 2-digit number. i 468 Select a number either 10 smaller or 10 larger. n 469 Find the middle number of the two (the average). g 470 Square this middle number (multiply it by itself). i 471 Subtract 25 from this square. n 3 Example: ? )
- 130. 472 If thefirst number is 36, choose 26 as the second number (10 smaller) . 479 S 473 The middle number (the average) is 31. u 474 Square this middle number: 31 x 31 = 961. (Remember how to square a 2-digit number b ending in 1?) tr 475 Subtract 25 from this square: 961 - 25 = 936 a (subtract mentally in steps: think 961 - 20 - 5 = 941 - 5 = 936). c t 476 So 36 x 26 = 936. 2 5 See the pattern? f r 477 If the first number is 78, you might pick 88 as the second numbe r (10 larger). o 478 The middle number (the average) is 83. m 3. Square this middle number: 83 x 83 = 6889. (Remember how to square a 2-digit number t his square: 6889 - 25 = 6864 (subtract mentally in steps : think 6889 - 20 - 5 = 6869 - 5 = 6864). 480 So 78 x 88 = 6864. Remember to subtract in easy steps and pick your number to get an easy square. With practice you will be a whiz at getting these products. Squaring numbers made up of nines 481 Choose a a number made up of nines (up to nine digits). 482 The answer will have one less 9 than the number, one 8, the same number of zeros as 9's, and a final 1 Example: 483 If the number to be squared is 9999 484 The square of the number has: one less nine than the number 9 9 9 one 8 8 the same number of zeros as 9's 0 0 0 a final 1 1
- 131. 3. So 9999x 9999 = 99980001. See the pattern? 485 If the number to be squared is 999999 486 The square of the number has: one less nine than the number 9 9 9 9 9 one 8 8 3 the same number of zeros as 9's 0 0 0 0 0 . a final 1 1 So 999999 x 999999 = 999998000001. This is not a very demanding mental math exercise, but it is an interesting pattern. Squaring numbers made up of ones 487 Choose a a number made up of ones (up to nine digits). 488 The answer will be a series of consecutive digits beginning with 1, up to the number of ones in the given number, and back to 1. Example: 489 If the number is 11111, (5 digits) -490 The square of the number is 23454321. (Begin with 1, up to 5, then back to 1.) See the pattern? 491 If the number is 1111111, (7 digits) -492 The square of the number is 1234567654321. (Begin with 1, up to 7, then back to 1.). This is an easy one, but it should be good for a quick example of your mental math abilities. Challenge a friend and BEAT THE CALCULATOR! Reversing/adding/subtracting 3-digit numbers 493 Select a number with different 1st and 3rd digits. 494 Reverse the digits. 495 Subtract the smaller digit from the larger. 496 Reverse the
- 132. digits of thisanswer. 497 Add these two numbers. 498 Subtract a number from this sum. 499 The final answer will be 1089 minus whatever number you choose in step 6! Example: 500 F o r t h e l a s t s tep you say to subtract 22. 501 The answer is : 1089 - 22 = 1067. See the pattern? 502 For the last step you say to subtract 101. 503 The answer is: 1089 - 101 = 988. Each time, change the number you select for the last step – just select one that is easy to subtract from 1089. Then your answers will be immediate and correct. Squaring numbers made up of sixes 504 Choose a a number made up of sixes. 2. The square is made up of: a. one fewer 4 than there are repeating 6's b. 3 c. same number of 5's as 4's d. 6 Example: 505 If the number to be squared is 666 506 The square of the number has: 4's (one less than digits in number) 4 4 3 3 5's (same number as 4's) 5 5 6 6 3. So 666 x 3666333 = 443556. See the pattern? 507 If the number to be squared is 66666 The square of the number has: 4's (one less than digits
- 133. in number) 44 4 4 3 3 5's (same number as 4's) 5 5 5 5 6 6 3. So 66666 x 66666 = 4444355556. Squaring a 2-digit number beginning with 1 508 Take a 2-digit number beginning with 1. 509 Square the second digit (keep the carry) _ _ X 510 Multiply the second digit by 2 and add the carry (keep the carry) _X_ 511 The first digit is one (plus the carry) X _ _ Example: 512 If the number is 16, square the second digit: 6 x 6 = 36 _ _ 6 513 Multiply the second digit by 2 and add the carry: 2 x 6 + 3 = 15 _ 5 _ 514 The first digit is one plus the carry: 1 + 1 = 2 2 _ _ 515 So 16 x 16 = 256. See the pattern? 516 For 19 x 19, square the second digit: 9 x 9 = 81 _ _ 1 518 Multiply the second digit by 2 and add the carry: 2 x 9 + 8 = 26 _ 6 _ The first digit is one plus the carry: 1 + 2 = 3 3 _ _ So 19 x 19 = 361. Finding the square root of 2-digit numbers ending in 1 519 Select a 2-digit number and square it. 520 Drop the last two digits of the square. 521 Find the largest square root of the remaining digits. This is first digit of the square root. 517 522 T h e s e c o n d digit is 1. Examples:
- 134. 523 If thesquare is 2601:524 Drop the last two digits: 26 525 Find the largest root in 26: 5 x 5 = 25 526 The first digit is 5. The second digit is 1. 527 So the square root of 2601 is 51. See the pattern? 528 If the square is 8281:529 Drop the last two digits: 82 530 Find the argest root in 82: 9 x 9 = 81 531 The first digit is 9. The second digit is 1. 532 So the square root of 2601 is 91. 533 534 This process also works for squares of 3-digit numbers (or more). For the square 22801, find the largest root of 228: 15x15 = 225. The first two digits are 15, the last digit is 1, and the square root of 22801 is 151. Finding the square root of perfect squares 548 S ending in 5 o t 535 Select a 2-digit number and square it. h 536 Drop the last two digits of the square. e 537 Find the largest square root of the remaining digits. s This is first digit of the square root. q u 538 The second digit is 5. a r Examples: e r 539 If the square is 9025: o 540 Drop the last two digits: 90 o 541 Find the largest root in 90: 9 x 9 = 81 t 542 The first digit is 9. The second digit is 5. o 543 So the square root of 9025 is 95. f 4 See the pattern? 2 2 544 If the square is 4225: 5 545 Drop the last two digits: 42 i 546 Find
- 135. the largest rootin 42: 6 x 6 = 36 s 547 The first digit is 6. The second digit is 5. 6 5 . This process also works for squares of 3-digit numbers (or more). For the square 15625, find the largest root of 156: 12x12 = 144 (not 13x13 = 169). The first two digits are 12, the last digit is 5, and the square root of 15625 is 125. Squaring special numbers (1 and repeating 3's) 549 Choose a number with a 1 and repeating 3's. 2. The square is made up of: first digits: 1 & one fewer 7 than repeating 3's next digits: 6 & one fewer 8 than repeating 3's last digit: 9 Example: 550 If the number to be squared is 13333: 551 The square has: first digits: 1 and one fewer 7 than 3's 1 7 7 next digits: 6 and one fewer 8 than 3's 6 8 8 last digit: 9 9 3. So 1333 x 1333 = 1776889. See the pattern? 552 If the number to be squared is 133333: The square has: 3 first digits: 1 and one fewer . S 7 than 3's 1 7 7 7 7 o next digits: 6 and one fewer 1 3 8 than 3's 6 8 8 8 8 3 3 last digit: 9 9 3 3 x 133333 = 17777688889.
- 136. Squaring special numbers(1 and repeating 6's) 553 Choose a number with a 1 and repeating 6's. 2. The square is made up of: first digits: 2 & one fewer 7 than repeating 6's next digits: same number of 5's as repeating 6's last digit: 6 Example: 554 If the number to be squared is 1666: 555 The square has: first digits: 2 and one fewer 7 than 6's 2 7 7 next digits: same number of 5's as 6's 5 5 5 last digit: 6 6 3. So 1666 x 1666 = 2775556. See the pattern? 556 If the number to be squared is 166666: The square has: first digits: 2 and one fewer 7 than 6's 2 7 7 7 7 next digits: same number of 5's as 6's 5 5 5 5 5 last digit: 6 6 3. So 166666 x 166666 = 27777555556. Squaring special numbers (1 and repeating 9's) n 557 Choose a number with a 1 and repeating 9's. e 2. The square is made up of: x first digits: 3 & one fewer 9 than repeating 9's t digits: 6 & one fewer 0 than repeating 9's last digit: 1 Example: 558 If the number to be squared is 1999: 559 The square has: first digits: 3 and one fewer 9 than 9's 3 9 9 next digits: 6 and one fewer 0 than 9's 6 0 0 last digit: 1 1 3. So 1999 x 1999 = 3996001. See the pattern? 560 If the number to be squared is 199999: The square has: first digits: 3 and one fewer 9 than 9's 3
- 137. 9 9 99 next digits: 6 and one fewer 0 than 9's & 6 0 0 0 0 last digit: 1 1 3. So 199999 x 199999 = 39999600001. Squaring special numbers (2 and repeating 3's) 561 Choose a number with a 2 and repeating 3's. 2. The square is made up of: first digits: 5 & one fewer 4 than repeating 3's next digits: 2 & one fewer 8 than repeating 3's last digit: 9 Example: 562 If the number to be squared is 2333: 563 The square has: first digits: 5 and one fewer 4 than 3's 5 4 4 next digits: 2 and one fewer 8 than 3's 2 8 8 last digit: 9 9 3. So 2333 x 2333 = 5442889. 564 If the number to be squared is 233333: The square has: first digits: 5 and one fewer 4 than 3's 5 4 4 4 4 next digits: 2 and one fewer 8 than 3's 2 8 8 8 8 last digit: 9 9 3. So 233333 x 233333 = 54444288889. Squaring special numbers (2 and repeating 6's) 565 Choose a number with a 2 and repeating 6's. 2. The square is made up of: first digits: 7 & two fewer 1's than repeating 6's next digits: 07 & one fewer 5 than repeating 6's last digit: 6 Example: 566 If the number to be squared is 2666: 567 The square has: first digits: 7 and two fewer 1's than 6's 7 1 next digits: 07 and one fewer 5 than 6's 0 7 5 5 last digit: 6 6 3. So 2666 x 2666 = 7,107,556. See the pattern?
- 138. 568 If thenumber to be squared is 266666: The square has: first digits: 7 and two fewer 's than 6's 7 1 1 1 next digits: 07 and one fewer 5 than 6's 0 7 5 5 5 5 last digit: 6 6 3. So 266666 x 266666 = 71,110,755,556. Squaring special numbers (2 and repeating 9's) 569 Choose a number with a 2 and repeating 9's. 2. The square is made up of: first digits: 8 & one fewer 9 than repeating 9's next digits: 4 & one fewer 0 than repeating 9's last digit: 1 Example: 570 If the number to be squared is 2999: 571 The square has: first digits: 8 and one fewer 9 than 9's 8 9 9 next digits: 4 and one fewer 0 than 9's 4 0 0 last digit: 1 1 3. So 2999 x 2999 = 8,994,001. See the pattern? 9 If the number to be squared is 299999: The square has: 3 first digits: 8 and one fewer . S 9 than 9's 8 9 9 9 9 o next digits: 4 and one fewer 2 9 0 than 9's 4 0 0 0 0 9 9 last digit: 1 1 9 9 x 299999 = 89,999,400,001. Squaring special numbers (3 and repeating 6's) Choose a number with a 3 and repeating 6's. 2. The square is made up of: first digits: 13 & two fewer 4's than repe
- 139. ating 6's nextdigits: 39 & one fewer 5 than repeating 6's last digit: 6 Example: If the number to be squared is 3666: The square has: first digits: 13 and two fewer 4's than 6's 1 3 4 next digits: 39 and one fewer 5 than 6's 3 9 5 5 last digit: 6 6 3. So 3666 x 3666 = 13,439,556. See the pattern? If the number to be squared is 366666: The square has: first digits: 13 and two fewer 4's than 6's 1 3 4 4 4 next digits: 39 and one fewer 5 than 6's 3 9 5 5 5 5 last digit: 6 6 3. So 366666 x 366666 = 134,443,955,556. Squaring special numbers (3 and repeating 9's) Choose a number with a 3 and repeating 9's. . The square is made up of: first digits: 15 & one fewer 9 than repeating 9's next digits: 2 & one fewer 0 than repeating 9's last digit: 1 Example: If the number to be squared is 3999: The square has: first digits: 15 and one fewer 9 than 9's 1 5 9 9 next digits: 2 and one fewer 0 than 9's 2 0 0 last digit: 1 1 3. So 3999 x 3999 = 15,992,001. See the pattern? If the number to be squared is 399999: The square has: first digits: 15 and one fewer 9 than 9's 1 5 9 9 9 9 next digits: 2 and one fewer 0 than 9's 2 0 0 0 0
- 140. last digit: 11 3. So 399999 x 399999 = 159,999,200,001. Squaring special numbers (4 and repeating 3's) Choose a number with a 4 and repeating 3's. E 2. The square is made up of: first digits: 18 & one fewer 7 than repeating 3's a next digits: 4 & one fewer 8 than repeating 3's m last digit: 9 p le: If the number to be squared is 4333: The square has: first digits: 18 and one fewer 7 than 3's 1 8 7 7 next digits: 4 and one fewer 8 than 3's 4 8 8 last digit: 9 9 3. So 4333 x 4333 = 18,774,889. See the pattern? f the number to be squared is 433333: The square has: first digits: 18 and one fewer 7 than 3's 1 8 7 7 7 7 next digits: 4 and one fewer 8 than 3's 4 8 8 8 8 last digit: 9 9 3. So 433333 x 433333 = 187,777,488,889. Squaring special numbers (4 and repeating 6's) Choose a number with a 4 and repeating 6's. 2. The square is made up of: first digits: 21 & one fewer 7 than repeating 6's next digits: 1 & one fewer 5 than repeating 6's last digit: 6 Example: If the number to be squared is 4666: The square has: first digits: 21 and one fewer 7 than 6's 2 1 7 7 next digits: 1 and one fewer 5 than 6's 1 :66 3. So 4666 x 4666 = 21,771,556.
- 141. See the pattern? Ifthe number to be squared is 466666: The square has: first digits: 21 and one fewer 7 than 6's 2 1 7 7 7 7 next digits: 1 and one fewer 5 than 6's 1 5 5 5 5 last digit: 6 6 3. So 466666 x 466666 = 217,777,155,556. Squaring special numbers (4 and repeating 9's) S Choose a number with a 4 and repeating 9's. e 2. The square is made up of: e first digits: 24 & one fewer 9 than Repeating 9's in the number tnext digits: same number of 0's as repeating 9's in the number hlast digit: 1 e Example: p at If the number to be squared is 4999: tThe square has: er first digits: 24 and one fewer 9 than 9's (in n the number) 2 4 9 9 next digits: same number ? of 0's as 9's (in the number) 0 0 0 last digit: 1 3. So 4999 x 4999 = 24,990,001. If the number to be squared is 499999: The square has: first digits: 24 and one fewer 9 than 9's 2 4 9 9 9 9 next digits: same number of 0's as 9's 0 0 0 0 0 last digit: 1 1 3. So 499999 x 499999 = 249,999,000,001. Squaring special numbers (5 and repeating 3's)
- 142. Choose a numberwith a 5 and repeating 3's. 2. The square is made up of: first digits: 28 & one fewer 4 than repeating 3's next digits: 0 & one fewer 8 than repeating 3's last digit: 9 Example: If the number to be squared is 5333: The square has: first digits: 28 and one fewer 4 than 3's 2 8 4 4 next digits: 0 and one fewer 8 than 3's 0 8 8 last digit: 9 9 3. So 5333 x 5333 = 28,440,889. See the pattern? If the number to be squared is 533333: The square has: first digits: 28 and one fewer 4 than 3's 2 8 4 4 4 4 next digits: 0 and one fewer 3's 0 8 8 8 8 last digit: 9 9 3. So 533333 x 533333 = 284,444,088,889. Squaring special numbers (5 and repeating 6's) Choose a number with a 5 and repeating 6's. 2. The square is made up of: first digits: 32 & two fewer 1's than repeating 6's next digits: 03 & one fewer 5 than repeating 6's last digit: 6 Example: If the number to be squared is 5666: The square has: first digits: 32 and two fewer 1's than 6's 3 2 1 next digits: 03 and one fewer 5 than 6's 0 3 5 5 last digit: 6 6 3. So 5666 x 5666 = 32,103,556. See the pattern? If the number to be squared is 566666: The square has: first digits: 32 and two fewer 1's than 6's 3 2 1 1 1 next digits: 03 and one fewer
- 143. 1 than 6's0 3 5 5 5 5 last digit: 6 6 3. So 566666 x 566666 = 321,110,355,556. Squaring special numbers (5 and repeating 9's) Choose a number with a 5 and repeating 9's. 2. The square is made up of: first digits: 35 & two fewer 9's than repeating 9's next digits: 88 & one fewer 0 than repeating 9's last digit: 1 Example: If the number to be squared is 5999: The square has: first digits: 35 and two fewer 9's than 9's 3 5 9 next digits: 88 and one fewer 0 than 9's 8 8 0 0 last digit: 1 1 3. So 5999 x 5999 = 35,988,001. See the pattern? If the number to be squared is 599999: The square has: first digits: 35 and two fewer 9's than 9's 3 5 9 9 9 next digits: 88 and one fewer 0 than 9's 8 8 0 0 0 0 last digit: 1 1 3. So 599999 x 599999 = 359,998,800,001. Squaring special numbers (6 and repeating 3's) Choose a number with a 6 and repeating 3's. f 2. The square is made up of: t first digits: 40 & two fewer 1's than repeating 3's h next digits: 06 & one fewer 8 than repeating 3's e last digit: 9 n u Example: mber to be squared is 6333: The square has: first digits: 40 and two fewer 1's than 3's 4 0
- 144. 1 next digits:06 and one fewer 8 than 3's 0 6 8 8 last digit: 9 9 3. So 6333 x 6333 = 40,106,889. See the pattern? If the number to be squared is 633333: The square has: first digits: 40 and two fewer 1's than 3's 4 0 1 1 1 next digits: 06 and one fewer 8 than 3's 0 6 8 8 8 8 last digit: 9 9 3. So 633333 x 633333 = 401,110,688,889. Squaring special numbers (6 and repeating 9's) S Choose a number with a 6 and repeating 9's. e 2. The square is made up of: e first digits: 48 & two fewer 9's than repeating 9's in the number tnext digits: 86 & one fewer 0 than repeating 9's in the number hlast digit: 1 e Example: p at If the number to be squared is 6999: tThe square has: er first digits: 48 and two fewer 9's than rep. n 9's 4 8 9 next digits: 86 and one fewer 0 ? than rep. 9's 8 6 0 0 last digit: 1 1 3. So 6999 x 6999 = 48,986,001. f the number to be squared is 699999: The square has: first digits: 48 and two fewer 9's than rep. 9's 4 8 9 9 9 next digits: 86 and one fewer 0 than rep. 9's 8 6 0 0 0 0
- 145. last digit: 11 3. So 699999 x 699999 = 489,998,600,001. Squaring special numbers (7 and repeating 3's) Choose a number with a 7 and repeating 3's. 2. The square is made up of: first digits: 53 & one fewer 7 than repeating 3's next digits: 2 & one fewer 8 than repeating 3's last digit: 9 Example: If the number to be squared is 7333: The square has: first digits: 53 and one fewer 7 than 3's 5 3 7 7 next digits: 2 and one fewer 8 than 3's 2 8 8 last digit: 9 9 3. So 7333 x 7333 = 53,772,889. See the pattern? If the number to be squared is 733333: The square has: igit: 9 3. So 733333 x 733333 = 537,777,288,889. Squaring special numbers (7 and repeating 6's) Choose a number with a 7 and repeating 6's. 2. The square is made up of: first digits: 58 & two fewer 7's than repeating 6's next digits: 67 & one fewer 5 than repeating 6's last digit: 6 Example: If the number to be squared is 7666: The square has: first digits: 58 and two fewer 7's than 6's 5 8 7 next digits: 67 and one fewer 5 than 6's 6 7 5 5 last digit: 6 6 3. So 7666 x 7666 = 58,767,556. See the pattern? first digits: 53 and one fewer 7 than 3's 5 3 7 7 7 7
- 146. next digits: 2and one fewer 8 than 3's 2 8 8 8 8 If the number to be squared is 766666: The square has: first digits: 58 and two fewer than 6's 5 8 7 7 7 next digits: 67 and one fewer 5 than 6's 6 7 5 5 5 5 last digit: 6 6 3. So 766666 x 766666 = 587,776,755,556. Squaring special numbers (7 and repeating 9's) Choose a number with a 7 and repeating 9's (use a mimimum of three 9's). 2. The square is made up of: first digits: 63 & two fewer 9's than repeating 9's in the number next digits: 84 & one fewe r 0 than repeating 9's in the number last digit: 1 Example: If the number to be squared is 7999: The square has: first digits: 63 and two fewer 9's than rep. 9's 6 3 9 next digits: 84 and one fewer 0 than rep. 9's 8 4 0 0 last digit: 1 1 3. So 7999 x 7999 = 63,984,001. See the pattern? If the number to be squared is 799999: The square has: 3 first digits: 63 and two fewer . S 9's than rep. 9's 6 3 9 9 9 o next digits: 84 and one fewer 7 9 0 than rep. 9's 8 4 0 0 0 0 9 9 last digit: 1 1 9 9 x 799999 = 639,998,400,001. Squaring special numbers (8 and repeating 3's) Choose a number with an 8 and (at least 3) repeating 3's.
- 147. 2. The squareis made up of: first digits: 69 & two fewer 4's than repeating 3's next digits: 3 & same number of 8's as repeating 3's last digit: 9 Example: If the number to be squared is 8333: The square has: first digits: 60 and two fewer 4's than 3's 6 9 4 next digits: 3 and same number of 8's as 3's 3 8 8 8 last digit: 9 9 3. So 8333 x 8333 = 69,438,889. See the pattern? If the number to be squared is 833333: The square has: first digits: 69 and two fewer 4's than 3's 6 9 4 4 4 next digits: 3 and same number of 8's as 3's 3 8 8 8 8 8 last digit: 9 9 3. So 833333 x 833333 = 694,443,888,889. Squaring special numbers (8 and repeating 9's) Choose a number with an 8 and repeating 9's (use a mimimum of three 9's). a 2. The square is made up of: s first digits: 80 & two fewer 9's tha n repeating 9's in the number t next digits: 82 & one fewer 0 than repeating 9's in the number d igit: 1 Example: If the number to be squared is 8999: The square has: first digits: 80 and two fewer 9's than rep. 9's 8 0 9 next digits: 82 and one fewer 0 than rep. 9's 8 2 0 0 last digit: 1 1 3. So 8999 x 8999 = 80,982,001. See the pattern? If the number to be squared is 899999: The square has: first digits: 80 and two fewer
- 148. 9's than rep.9's 8 0 9 9 9 next digits: 82 and one fewer 0 than rep. 9's 8 2 0 0 0 0 last digit: 1 1 3. So 899999 x 899999 = 809,998,200,001. Squaring special numbers (9 and repeating 3's) Choose a number with a 9 and repeating 3's (use at least three 3's). 2. The square is made up of: first digits: 87 & two fewer 1's than repeating 3's next digits: 04 & one fewer 8 than repeating 3's last digit: 9 Example: If the number to be squared is 9333: The square has: : 87 and two fewer 1's than 3's 8 7 1 next digits: 04 and one fewer 8 than 4's 0 4 8 8 last digit: 9 9 3. So 9333 x 9333 = 87,104,889. See the pattern? If the number to be squared is 933333: ihe square has: first digits: 87 and two fewer 1's than 3's 8 7 1 1 1 next digits: 04 and one fewer 8 than 3's 0 4 8 8 8 8 last digit: 9 9 3. So 933333 x 933333 = 871,110,488,889. Squaring special numbers (9 and repeating 6's) Choose a number with a 9 and repeating 6's (use at least three 6's). 2. The square is made up of: first digits: 93 & two fewer 4's than repeating 6's next digits: 31 & one fewer 5 than repeating 6's last digit: 6 Example: If the number to be squared is 9666: The square has:
- 149. first digits: 93and two fewer 4's than 6's 9 3 4 next digits: 31 and one fewer 5 than 6's 3 1 5 5 last digit: 6 6 3. So 9666 x 9666 = 93,431,556. See the pattern? If the number to be squared is 966666: The square has: first digits: 93 and two fewer 4's than 6's 9 3 4 4 4 next digits: 31 and one fewer 5 than 6's 3 1 5 5 5 5 last digit: 6 6 3. So 966666 x 966666 = 934,443,155,556. Squaring numbers in the 20s Square the last digit (keep the carry) _ _ X Multiply the last digit by 4, add the carry _ X _ The first digit will be 4 plus the carry: X _ _ Example: If the number to be squared is 24: Square the last digit (keep the carry): 4 x 4 = 16 (keep 1) _ _ 6 Multiply the last digit by 4, add the carry: 4 x 4 = 16, 16 + 1 = 17 _ 7 _ The first digit will be 4 plus the carry: 4 (+ carry): 4 + 1 = 5 5 _ _ So 24 x 24 = 576. See the pattern? If the number to be squared is 26: Square the last digit (keep the carry): 6 x 6 = 36 (keep 3) _ _ 6 Multiply the last digit by 4, add the carry: 4 x 6 = 24, 24 + 3 = 27 (keep 2) _ 7 _ The first digit will be 4 plus the carry: 4 (+ carry): 4 + 2 = 6 6 _ _. So 26 x 26 = 676. Squaring numbers in the 30s
- 150. Square the lastdigit (keep the carry) _ _ _ X Multiply the last digit by 6, add the carry _ _ X _ The first digits will be 9 plus the carry: X X _ _ Example: If the number to be squared is 34: Square the last digit (keep the carry): 4 x 4 = 16 (keep 1) _ _ _ 6 Multiply the last digit by 6, add the carry: 6 x 4 = 24, 24 + 1 = 25 _ _ 5 _ The first digits will be 4 plus the carry: 9 (+ carry): 9 + 2 = 11 11__ So 34 x 34 = 1156. See the pattern? If the number to be squared is 36: Square the last digit (keep the carry): 6 x 6 = 36 (keep 3) _ _ _ 6 Multiply the last digit by 6, add the carry: 6 x 6 = 36, 36 + 3 = 39 (keep 3) _ _ 9 _ The first digits will be 9 plus the carry: 9 (+ carry): 9 + 3 = 12 1 2 _ _. So 36 x 36 = 1296. Squaring numbers in the 40s Square the last digit (keep the carry) _ _ X Multiply the last digit by 8, add the carry _ X _ f The first digits will be 16 plus the carry: X X _ _ t Example: h e number to be squared is 42: Square the last digit: 2 x 2 = 4 _ _ _ 4 Multiply the last digit by 8: 8 x 2 = 16 _ _ 6 _ The first digits will be 16 plus the carry: 16 (+ carry): 16 + 1 = 17 1 7 _ _ So 42 x 42 = 1764.
- 151. See the pattern? Ifthe number to be squared is 48: Square the last digit (keep the carry): 8 x 8 = 64 (keep 6) _ _ _ 4 Multiply the last digit by 8, add the carry: 8 x 8 = 64, 64 + 6 = 70 (keep 7) _ _ 0 _ The first digits will be 16 plus the carry: 16 (+ carry): 16 + 7 = 23 2 3 _ _ So 48 x 48 = 2304. Squaring numbers in the 50s 2 Square the last digit (keep the carry) _ _ _ X 5 Multiply the last digit by 10, add the carry _ _ X _ (The first digits will be 25 plus t he carry: X X _ _ + c Example:a r If the number to be squared is 53: r y Square the last digit (keep the carry): ): 3 x 3 = 9 (keep 3) _ _ _ 9 2 5 Multiply the last digit by 10, add the carry: + 10 x 3 = 30 (keep 3) _ _ 0 _ 3 = The first digits will be 25 plus the carry: 2 8 2 8 _ _ So 53 x 53 = 2809. See the pattern? If the number to be squared is 56: Square the last digit (keep the carry): 6 x 6 = 36 (keep 3) _ _ _ 6 Multiply the last digit by 10, add the carry: 10 x 6 = 60, 60 + 3 = 63 _ _ 3 _ The first digits will be 25 plus the carry: 25 (+ carry): 25 + 6 = 31 3 1 _ _ So 53 x 53 = 3136. Squaring numbers in the 60s Square the last digit (keep the carry) _ _ _ X Multiply the last digit by 12, add the carry _ _ X _ The first digits will be 36 plus the carry: X X _ _
- 152. Example: If the numberto be squared is 63: Square the last digit (keep the carry): 3 x 3 = 9 (keep 3) _ _ _ 9 Multiply the last digit by 12, add the carry: 12 x 3 = 36 (keep 3) _ _ 6 _ The first digits will be 36 plus the carry: 36 (+ carry): 36 + 3 = 39 3 9 _ _ So 63 x 63 = 3969. See the pattern? If the number to be squared is 67: Square the last digit (keep the carry): 7 x 7 = 49 (keep 4) _ _ _ 9 Multiply the last digit by 12, add the carry: 12 x 7 = 84, 84 + 4 = 88 __8_ The first digits will be 36 plus the carry: 36 (+ carry): 36 + 8 = 44 4 4 _ _ So 67 x 67 = 4489. Squaring numbers in the 70s Square the last digit (keep the carry) _ _ _ X Multiply the last digit by 14, add the carry _ _ X _ The first digits will be 49 plus the carry: X X __ Example: If the number to be squared is 72: Square the last digit: 2 x 2 = 4 _ _ _ 4 Multiply the last digit by 14: 14 x 2 = 28 (keep the carry) _ _ 8 _ The first digits will be 49 plus the carry: 49 (+ carry): 49 + 2 = 51 5 1 _ _ So 72 x 72 = 5184. See the pattern? If the number to be squared is 78: Square the last digit (keep the carry): 8 x 8 = 64 (keep 6) _ _ _ 4
- 153. Multiply the lastdigit by 14, add the carry: 14 x 8 = 80 + 32 = 112 112 + 6 = 118 (keep 11) _ _ 8 _ The first digits will be 49 plus the carry (11): 49 (+ carry): 49 + 11 = 60 6 0 _ _. So 78 x 78 = 6084. Squaring numbers in the 80s Square the last digit (keep the carry) _ _ X Multiply the last digit by 16, add the carry _ X _ The first digits will be 64 plus the carry: X X _ _ Example: If the number to be squared is 83: Square the last digit: 3 x 3 = 9 _ _ _ 9 Multiply the last digit by 16: 16 x 3 = 30 + 18 = 48 _ _ 8 _ The first digits will be 64 plus the carry: 64 (+ carry): 64 + 4 = 68 6 8 _ _ So 83 x 83 = 6889. See the pattern? If the number to be squared is 86: Square the last digit (keep the carry): 6 x 6 = 36 (keep 3) _ _ _ 6 Multiply the last digit by 16, add the carry: 16 x 6 = 60 + 36 = 96 96 + 3 = 99 (keep 9) _ _ 9 _ The first digits will be 64 plus the carry: 64 (+ carry): 64 + 9 = 73 7 3 _ _ So 86 x 86 = 7396. Squaring numbers in the 200s T hen Choose a number in the 200s (practice with numbers under 210 , then progress to larger ones). e The first digit of the square is 4: 4 _ _ _ _
- 154. t two digitswill be 4 times the last 2 digits: _ X X _ _ The last two places will be the square of the last digit: _ _ _ X X Example: If the number to be squared is 206: The first digit is 4: 4 _ _ _ _ The next two digits are 4 times the last digit: 4 x 6 = 24: _ 2 4__ Square the last digit: 6 x 6 = 36: _ _ _ 3 6 So 206 x 206 = 42436. For larger numbers work right to left: Square the last two digits (keep the carry): _ _ _ X X 4 times the last two digits + carry: _ X X _ _ Square the first digit + carry: X____ See the pattern? If the number to be squared is 225: Square last two digits (keep carry): 25x25 = 625 (keep 6): _ _ _ 2 5 4 times the last two digits + carry: 4x25 = 100; 100+6 = 106 (keep 1): _ 0 6 _ _ Square the f irst digit + carry: 2x2 = 4; 4+1 = 5: 5 _ _ _ _ So 225 x 225 = 50625. Squaring numbers in the 300s S q Choose a number in the 300s (practice with numbers under 310, then progress to larger ones). u The first digit of the square is 9: 9 _ _ _ _ a The next two digits will be 6 times the last 2 digits: _ X X _ _ r The last two places will be the square of the last digit: _ _ _ X X e t Example: h e If the number to be squared is 309: The first digit is 9: 9 _ _ _ _a The next two digits are 6 times the last digit: s 6 x 9 = 54: _ 5 4
- 155. _ _ tdigit: 9 x 9 = 81: _ _ _ 8 1 So 309 x 309 = 95481. For larger numbers reverse the steps: Square the last two digits (keep the carry): _ _ _ X X 6 times the last two digits + carry: _ X X _ _ Square the first digit + carry: X____ See the pattern? If the number to be squared is 325: Square last two digits (keep carry): 25x25 = 625 (keep 6): _ _ _ 2 5 6 times the las t two digits + carry: 6x25 = 150; 150+6 = 156 (keep 1): _ 5 6 _ _ Square the first digit + carry: 3x3 = 9; 9+1 = 10: 1 0 _ _ _ _ So 325 x 325 = 105625. Squaring numbers in the 400s S Choose a number in the 400s (keep the numbers low at first; then progress to larger ones). qThe first two digits of the square are 16: 1 6 _ _ _ _ u The next two digits will be 8 times the last 2 digits: _ _ X X _ _ a The last two places will be the square of the last two digits: _ _ _ _ X X r e Example: t h If the number to be squared is 407: e The first two digits are 16: 1 6 _ _ _ _ l The next two digits are 8 times the last 2 digits: a 8 x 7 = 56: _ _ 5 6 _ _s t Square the last digit: 7 x 7 = 49: _ _ _ 4 9 t So 407 x 407 = 165,649. w o For larger numbers reverse the steps: d i
- 156. Square the lasttwo digits (keep the carry): _ _ _ _ X X g 8 times t he last two digits + carry: _ _ X X _ _ it 16 + carry: X X _ _ _ _ s (k See the pattern? e If the number to be squared is 425: p e the carry): 25 x 25 = 625 (keep 6): _ _ _ _ 2 5 8 times the last two digits + carry: 8 x 25 = 200; 200 + 6 = 206 (keep 2): _ _ 0 6 _ _ 16 + carry: 16 + 2 = 18: 1 8 _ _ _ _ So 425 x 425 = 180,625. Squaring numbers in the 500s Choose a number in the 500s (start with low numbers at first; then graduate to larger ones). The first two digits of the square are 25: 2 5 _ _ _ _ The next two digits will be 10 times the last 2 digits: _ _ X X _ _ The last two places will be the square of the last two digits: _ _ _ _ X X Example: If the number to be squared is 508: The first two digits are 25: 25_ _ _ _ The next two digits are 10 times the last 2 digits: 10 x 8 = 80: __80__ Square the last digit: 8 x 8 = 64: _ _ _ 6 4 So 508 x 508 = 258,064. For larger numbers reverse the steps: Square the last two digits (keep the carry): _ _ _ _ X X 10 times the last two digits + carry: _ _ X X _ _ 25 + carry: X X _ _ _ _
- 157. If the numberto be squared is 525: Square the last two digits (keep the carry): 25 x 25 = 625 (keep 6): _ _ _ _ 2 5 10 times the last two d igits + carry: 10 x 25 = 250; 250 + 6 = 256 (keep 2): _ _ 5 6 _ _ 25 + carry: 25 + 2 - 27: 2 7 _ _ _ _ Squaring numbers in the 600s Choose a number in the 600s (practice with smaller numbers, then progress to larger ones). S The first two digits of the square are 36 : 3 6 _ _ _ _ qThe next two digits will be 12 times the last 2 digits: _ _ X X _ _ u The last two places will be the square of the last two digits :____XXa r Example: e t If the number to be squared is 607: h The first two digits are 36: 3 6 _ _ _ _ e The next two digits are 12 times the last 2 digits: 12 x 07 = 84: _ _ 8 4 _ _ a st 2 digits: 7 x 7 = 49: _ _ _ _ 4 9 So 607 x 607 = 368,449. For larger numbers reverse the steps: If the number to be squared is 625: Square the last two digits (keep carry) : 25x25 = 625 (keep 6): _ _ _ _ 2 5 12 times the last 2 digits + carry: 12x25 = 250 + 50 = 300 + 6 = 306: _ _ 0 6 _ _ 36 + carry: 36 + 3 = 39: 3 9 _ _ _ _ 5. So 625 x 625 = 390,625. Squaring numbers in the 700s 3 Choose a number in the 700s (practice with smaller numbers, then
- 158. progress to largerones). . Square the last two digits (keep the carry): _ _ _ _ X X Multiply the last two digits by 14 and M add the carry: _ _ X X__u lt The first two digits will be 49 plus the carry: X X _ _ _ _ i pExample: l y If the number to be squared is 704: t Square the last two digits (keep t he carry): h 4 x 4 = 16: _ _ _ _ 1 6 e l Multiply the last two digits by 14 and a add the carry: 14 x 4 = 56: _ _ 5 6__s t The first two digits will be 49 plus the carry: 4 9 _ _ _ _ t So 704 x 70 4 = 495,616. w o See the pattern? d i If the number to be squared is 725: g Square the last two digits (keep t he carry): it 25 x 25 = 625: _ _ _ _ 2 5 s b y 14 and add the carry: 14 x 25 = 10 x 25 + 4 x 25 = 250 + 100 = 350. 35 0+6= 356: 56: _ _ 5 6 _ _ The first two digits will be 49 plus the carry: 49 + 3 = 52: 5 2 _ _ _ _ So 725 x 725 = 525,625. Squaring numbers between 800 and 810 Choose a number between 800 and 810. Square the last two digits: _ ___XX Multiply the last two digits by 16
- 159. (keep the carry):_ _ X X _ _ Square 8, add the carry: X X _ _ _ _ Example: If the number to be squared is 802: Square the last two digits: 2 x 2 = 4: _ _ _ _ 0 4 Multiply the last two digits by 16: 16 x 2 = 32: _ _ 3 2 _ _ Square 8: 6 4 _ _ _ _ So 802 x 802 = 643,204. See the pattern? If the number to be squared is 807: Square the last two digits: 7 x 7 = 49: _ _ _ _ 4 9 Multiply the last two digits by 16 (keep the carry): 16 x 7 = 112: _ _ 1 2 _ _ Square 8, add the carry (1): 6 5 _ _ _ _ So 807 x 807 = 651, 249. Squaring numbers in the 900s e Choose a number in the 900s - start out easy with numbers near 1000 ; then go lower when pert. Subtract the number from 1000 to get the difference. The first three places will be the number minus the difference: X X X _ _ _. The last three places will be the square of the difference: _ _ _XX X (if 4 digits, add the first digit as carry). Example: If the number to be squared is 985: Subtract 1000 - 985 = 15 (differe nce) Number - difference: 985 - 15 = 970: 9 7 0 _ _ _ Square the difference: 15 x 15 = 225: _ _ _ 2 2 5 So 985 x 985 = 970225. See the pattern?
- 160. If the numberto be squared is 920: Subtract 1000 - 920 = 80 (difference) Number - difference: 920 - 80 = 840: 8 4 0 _ _ _ Square the difference: 80 x 80 = 6400: _ _ _ 4 0 0 Carry first digit when four digits: 8 4 6 _ _ _ 6. So 920 x 920 = 846400. Squaring numbers between 1000 and 1100 Choose a number between 1000 and 1100. S The first two digits are: 1,0 _ _, _ _ _ qFind the difference between your number and 1000. u Multiply the difference by 2: 1,0 X X, _ _ _ a Square the difference: 1,0 _ _, X X X r e Example: t h If the number to be squared is 1007: e The first two digits are: 1,0 _ _, _ _ _ d Find the difference: 1007 - 1000 = 7 if Two times the difference: 2 x 7 = 14: f 1,0 1 4, _ _ _ e rence: 7 x 7 = 49: 1,0 1 4, 0 4 9 So 1007 x 1007 = 1,014,049. See the pattern? If the number to be squared is 1012: The first two digits are: 1,0 _ _, _ _ _ Find the difference: 1012 - 1000 = 12 Two times the difference: 2 x 12 = 24: 1,0 2 4, _ _ _ Square the difference: 12 x 12 = 144: 1,0 2 4, 1 4 4 So 1012 x 1012 = 1,024,144. Start with lower numbers and then extend your expertise to all the numbers between 1000 and 1100. Remember to add the first digit as carry when the square of the difference is four digits. Squaring numbers between 2000 and 2099 For
- 161. Choose a numberbetween 2000 and 2099. (Start with numbers below 2025 to begin with, then graduate to larger numbers.) l The first two digits are: 4 0 _ _ _ _ _ a The next two digits are 4 times the last two digits: r 4 0 X X _ _ _ g e For the last three digits, square the last two digits in the number chosen (insert zeros when r needed): 4 0 _ _ X X X n Example: u m If the number to be squared is 2003: b The first two digits are: 4 0 _ _ _ _ _ e The next two digits are 4 times the last two: r 4 x3= 12: _ _ 1 2 _ _ _ s ,For the last three digits, square the last two: 3 x 3 = 9: _ _ _ _ 0 0 9 r e So 2003 x 2003 = 4,012,009. v e See the pattern? r se the order: If the number to be squared is 2025: For the last three digits, square the last two: 25 x 25 = 625: _ __ _ 6 2 5 The middle two digits are 4 times the last two (keep the carry): 4 x 25 = 100 (keep carry of 1): _ _ 0 0 _ The first two digits are 40 + the carry: 40 + 1 = 41: 4 1 _ _ _ _ _ So 2025 x 2025 = 4,100,625. Sing numbers between 3000 and 3099 I f t h
- 162. Choose a numberbetween 3000 and 3099. (Start with numbers below 3025 to begin with, then e graduate to larger numbers.) n The first two digits are: 9 0 _ _ _ _ _ u The next two digits are 6 times the last two digits: m 9 0 X X _ _ _ b e For the last three digits, square the last two digits in the number chosen (insert zeros when r needed): 9 0 _ _ X X X t o Example: b e If the number to be squared is 3004: s The first two digits are: 90_ _ _ _ _ q The next two digits are 6 times the last two: u 6 x 4 = 24: _ _ 2 4 _ _ _ a For the last three digits, square the last two: e 4 x 4 = 16: d i So 3004 x 3004 = 9,024,016. s 3 See the pattern? 0 25 For larger numbers, reverse the order: : For the last three digits, square the last two: T 25 x 25 = 625: __ __625h e The middle two digits are 6 times the last two (keep the carry): 6 x 25 = 150 (keep carry of 1): _ _ 5 0 _ fi _ _ r st two digits are 90 + the carry: 90 + 1 = 91: 9 1 _ _ _ _ _ So 3025 x 3025 = 9,150,625. Squaring numbers between 4000 and 4099 f t h
- 163. Choose a numberbetween 4000 and 4099. e For numbers less than 4013: n The first three digits are: 1 6 0 _ _ _ _ _ u The nex t two digits are 8 times the last two digits: m _ _ _ X X _ _ _ b e For the last three digits, square the last two digits in the numbe r chosen (insert zeros when r needed): _ _ _ _ _ X X X t o Example: b e If the number to be squared is 4005: s The first three digits are: 1 6 0 _ _ _ _ _ q The next two digits are 8 times the last two: u 8x 5 = 40: _ _ 4 0 _ _ _ a r For the last three digits, square the last two: e 5 x 5 = 25: _ _ _ __ 025d i So 4005 x 4005 = 16,040,025. s 4 See the pattern? 0 8 0 For numbers greater than 4012, reverse the order: : For the last three digits, square the last two: 80 x 80 = 6400, carry 6: _ _ _ _ 4 0 0 The middle two digits are 8 times the last two (keep the carry): 8 x 80 = 640 (keep carry of 6), 40 + 6: _ __46___ The first three digits are 160 + the carry: 160 + 6 = 166: 1 6 6 _ _ _ _ _ Squaring numbers between 5000 and 5099
- 164. T Choose anumber between 5000 and 5099. h The first three digits are: 2 5 0 _ _ _ _ _ e next two digits are 10 times the last two digits: 2 5 0 X X _ _ _ For the last three digits, square the last two digits in the number chosen (insert zeros when needed): 2 5 0 _ _ X X X Example: If the number to be squared is 5004: The first three digits are: 250_ _ _ _ _ The next two digits are 10 times the last two: 10 x 4 = 40: _ _ 4 0 _ _ _ For the last three digits, square the last two: 4 x 4 = 16: _ _ _ _ _ 0 1 6 So 5004 x 5004 = 25,040,016. See the pattern? For numbers greater than 5011, reverse the order: If the number to be squared is 5012: For the last three digits, square the last two: 12 x 12 = 144: _ _ _ _ 1 4 4 The middle two digits are 10 times the last two (keep the carry): 10 x 12 = 120 (keep carry of 1): _ _ _ 20___ The first three digits are 150 + the carry: 250 + 1 = 251: 2 5 1 _ _ _ _ _ So 5012 x 5012 = 25,120,144. Squaring numbers between 6000 and 6099 Choose a number between 6000 and 6099. f The first three digits are: 3 6 0 _ _ _ _ _ t The next two digits are 12 times the last two digits: h _ _ _ X X _ _ e n For the last three digits, square the last two digits in the number
- 165. chosen (insert zeroswhen u needed): _ _ _ _ _ X X X m b Example: e r to be squared is 6004: The first three digits are: 3 6 0 _ _ _ _ _ The next two digits are 12 times the last two: 12 x 4 = 48: _ _ _ 4 8 ___ For the last three digits, square the last two: 4 x 4 = 16: _ _ _ _ 0 1 6 So 6004 x 6004 = 36,048,016. See the pattern? For numbers greater than 6008, reverse the order: If the number to be squared is 6020: For the last three digits, square the last two: 20 x 20 = 400: _ _ _ _ 4 0 0 The middle two digits are 12 times the last two: 12 x 20 = 240 (keep carry): _ _ _ 4 0 _ _ _ The first digits are 360 + the carry: 360 + 2 = 362: 3 6 2 _ _ _ _ _ So 3025 x 3025 = 36,240,400. Squaring numbers between 7000 and 7099 4 Choose a number between 7000 and 7099. x The first three digits are : 4 9 0 _ _ _ _ _ 4 The next two digits are 4 times the last two digits, = with zero added: _ _ _ X X _ _ 1 6 For the last three digits, square the last two digits in the number chosen (insert zeros when ; needed): _ _ _ _ _ X X X 1 6 Example: + 4 If the number to be squared is 7004: 0 The first three digits are: 4 9 0 _ _ _ _ _ = The next two digits are 4 times the last two, with zero added: 5 6: _ _ _ 5 6 _ _ _
- 166. For the lastthree digits, square the last two: 4 x 4 = 16: _ _ _ _016 So 7004 x 7004 = 49,056,016. See the pattern? For numbers greater than 7007, reverse the order: If the number to be squared is 7025: For the last three digits, square the last two: 25 x 25 = 625: _ _ _ _ 6 2 5 For the middle two digits, add zero to the last two, then add 4 times the last two : 250 + 4 x 25: 250 + 100 = 350 (keep carry): _ _ _ 5 0 _ _ _ The first three digits are 490 + the carry: 490 + 3 = 493: 4 9 3 _ _ _ _ _ So 7025 x 7025 = 49,350,625. Squaring numbers in the hundreds S Choose a number over 100 (keep it low for practice, o then go higher when expert). 1 1 The last two places will be the square of 2 the last two digits (keep any carry) _ _ _ X X. x 1 The first three places will be the number plus 1 the last two digits plus any carry : X X X _ _. 2 = 1 Example: 2 5 If the number to be squared is 106: 4 Square the last two digits (no carry): 6 x 6 = 36: _ _ _ 3 6 4 Add the last two digits (06) to the number: 106 + 6 = 112: 1 1 2 _ _ So 106 x 106 = 11236. See the pattern? If the number to be squared is 112: Square the last two digits (keep carry 1): 12 x 12 = 144: _ _ _ 4 4 Add the last two digits (12) plus the carry (1) to the number: 112 + 12 + 1 = 125: 1 2 5 _ _ With a little practice your only limit will be your ability to square the last two digits!
- 167. Squaring a repeating6-digit number Choose a number with repeating 6's . 2. The square is made up of: One less 4 than there are digits in the number; One 3; The same number of 5's as 4's; A final 6. Example: If the number to be squared is 666: The square has: one less 4 than digits in the number 4 4 one 3 3 same number of 5's as 4's 5 5 a final 6 6 3. So the square of 666 is 443556. See the pattern? If the number to be squared is 66666: The square has: one less 4 than digits in the number) 4 4 4 4 one 3 3 same number of 5's as 4's 5 5 5 5 a final 6 6 3. So the square of 66666 is 4444355556. Squaring special numbers (3's and final 1) Choose a number with repeating 3's and a final 1. E 2. The square is made up of: x One fewer 1 than there are repeating 3's a 09 m The same number of 5's as there are 1 's in the square; pA final 61 le: If the number to be squared is 3331: The square has: Two 1's (one fewer than repeating 3's) 1 1 Next digits: 09 0 9 Two 5's (same as 1's in square) 5 5 A final 61 6 1 3. So the square of 3331 is 11,095,561. See the pattern? If the number to be squared is 333331: The square has: Four 1's (one fewer than repeating 3's) 1 1 1 1 Next digits: 09 0 9
- 168. Four 5's (sameas 1's in square) 5 5 5 5 A final 61 6 1 3. So the square of 333331 is 111,109,555,561. Squaring special numbers (3's and final 2) Choose a number with repeating 3's and a final 2. f 2. The square is made up of: t the same number of 1's as there are repeating 3's; h a zero; e the same number of 2 's as there are 1's in the square; n a final 4. u m Example: b e r to be squared is 3332: The square has: three 1's (number of 3's in number) 1 1 1 a zero 0 three 2's (same as 1's in square) 2 2 2 a final 4 4 3. So the square of 3332 is 11,102,224. See the pattern? If the number to be squared is 333332: The square has: five 1's (number of 3's in number) 1 1 1 1 1 a zero 0 five 2's (same as 1's in square) 2 2 2 2 2 a final 4 4 3. So the square of 333332 is 111,110,222,224. These big squares should be quite impressive, and difficult for others to check unless they have a huge calculator. Squaring special numbers (3's and final 4) S Choose a number with repeating 3's and a final 4. e 2. The square is made up of: e the same number of 1's as ther e are digits in the number; � one fewer 5; ta final 6 h e Example: p a If the number to be squared is 3334: tThe square has: t e four 1's (number of digits r
- 169. in number) 11 1 1 three 5's (one fewer) 5 5 5 a final 6 n ? 3. So the square of 3334 is 11,115,556. f the number to be squared is 333334: The square has: six 1's 1 1 1 1 1 1 five 5's 5 5 5 5 5 a final 6 6 3. So the square of 333334 is 111,111,555,556. Squaring special numbers (3's and final 5) T Choose a number with repeating 3's and a final 5. h 2. The square is made up of: e the same number of 1's as there are repeating 3's in the number; � one more 2 than there are s repeating 3's; qa final 5. u a Example: r e If the number to be squared is 3335: h The square has: a s three 1's (same as repeating 3's) 1 1 1 four : 2's (one more tha repeating 3's) 2 2 2 2 a final 5 5 3. So the square of 3335 is 11,122,225. See the pattern? If the number to be squared is 333335: S five 1's (same as repeating 3's) 1 1 1 1 1 qsix 2's (one more than repeating 3's) 2 2 2 2 2 2 a final 5 5 ar 3. So the square of 333335 is 111,112,222,225. n g special numbers (3's and final 6) Choose a number with repeating 3's and a final 6. 2. The square is made up of: the same number of 1's as there are repeating 3's in the number; � one 2 � one fewer 8 than there are repeating 3's; a final 96.
- 170. Example: If the numberto be squared is 3336: The square has: three 1's (same as repeating 3's) 1 1 1 one 2 2 two 8's (one fewer than repeating 3's) 8 8 a final 96 9 6 3. So the square of 3336 is 11,128,896. See the pattern? the number to be squared is 333336: The square has: five 1's (same as repeating 3's) 1 1 1 1 1 one 2 2 four 8's (one fewer than repeating 3's) 8 8 8 8 a final 96 9 6 3. So So 333336 x 3333336 = 111,112,888,896. Squaring special numbers (3's and final 7) Choose a number with repeating 3's and a final 7. 2. The square is made up of: the same number of 1's as there are repeating 3's in the number; � one 3 � one fewer 5 than there are repeating 3's; a final 69. Example: If the number to be squared is 3337: The square has: three 1's (same as repeating 3's) 1 1 1 one 3 3 two 5's (one fewer than repeating 3's) 5 5 a final 69 6 9 3. So the square of 3337 is 11,135,569. See the pattern? If the number to be squared is 333337: The square has: five 1's (same as
- 171. repeating 3's) 11 1 1 1 one 2 2 four 5's (one fewer than repeating 3's) 5 5 5 5 a final 69 6 9 3. So So 333337 x 3333337 = 111,113,555,569. Squaring special numbers (3's and final 8) ho ose a number with repeating 3's and a final 8. 2. The square is made up of: the same number of 1's as there are repeating 3's in the number; � one 4 � one fewer 2 than there are repeating 3's; a final 44. Example: If the number to be squared is 33338: The square has: four 1's (same as repeating 3's) 1 1 1 1 one 4 4 three 2's (one fewer than repeating 3's) 2 2 2 a final 44 4 4 3. So the square of 33338 is 1,111,422,244. See the pattern? If the number to be squared is 3333338: The square has: six 1's (same as repeating 3's) 1 1 1 1 1 1 one 4 4 five 2's (one fewer than g 3's) 2 2 2 2 2 a final 44 4 4 3. So So 3333338 x 3333338 = 11,111,142,222,244. Squaring special numbers (3's and final 9) 3. So the square of 3333339 = 11,111,148,888,921. Choose a number with repeating 6's and a final 1. 2. The square is made up of: � one fewer 4 than there are repeating 6's
- 172. 36 two fewer8's than there are repeating 6's A final 921 Example: If the number to be squared is 6661: The square has: two 4's (one fewer than repeating 6's) 4 4 Next digits: 36 3 6 one 8 (two fewer than repeating 6's) 8 A final 921 9 2 1 3. So the square of 6661 is 44,368,921. See the pattern? If the number to be squared is 666661: The square has: four 4's (one fewer than repeating 6's) 4 4 4 4 Next digits: 36 3 6 three 8's 8 8 8 A final 921 9 2 13. So the square of 666661 is 44,4436,888,921. Squaring special numbers (6's and final 2) s Choose a number with repeating 6's and a final 2. a 2. The square is made up of: � one fewer 4 than there are repeating 6's m 38 e number of 2's as 4's in the square a final 44 Example: If the number to be squared is 6662: The square has: two 4's (one fewer than repeating 6's) 4 4 Next digits: 38 3 8 two 2's (same number as repeating 6's) 2 2 A final 44 4 4 3. So the square of 6662 is 44,382,244. See the pattern? If the number to be squared is 666662: The square has: four 4's (one fewer than repeating 6's) 4 4 4 4 Next digits: 38 3 8 four 2's (same as repeating 6's) 2 2 2 2 A final 44 4 4 3. So the square of 666662 is 444,438,222,244. Squaring special numbers (6's and final 3)
- 173. I Choose anumber with repeating 6's and a final 3. f 2. The square is made up of: � one fewer 4 than there are repeating 6's t 39 h same number of 5's as 4's in the square e a final 69 n u Example: m ber to be squared is 6663: The square has: two 4's (one fewer than repeating 6's) 4 4 Next digits: 39 3 9 two 5's (same number as repeating 6's) 5 5 A final 69 6 9 3. So the square of 6663 is 44,395,569. See the pattern? If the number to be squared is 666663: The square has: four 4's (one fewer than repeating 6's) 4 4 4 4 Next digits: 39 3 9 four 5's (same as repeating 6's) 5 5 5 5 A final 69 6 9 3. So the square of 666663 is 444,439,555,569. Squaring special numbers (6's and final 4) t Choose a number with repeating 6's and a final 4. h 2. The square is made up of: r the same number of 4's as repeating 6's e 0 � one fewer 8 than repeating 6's e a final 96 ' Example: s If the number to be squared is 6664: s ( The square has: a m e number as repeating 6's) 4 4 4 next digit: 38 0 two 8's (one fewer than repeating 6's) 8 8 a final 96 9 6 3. So the square of 6664 is 44,408,896. See the pattern? If the number to be squared is 666664:
- 174. The square has: five4's (same number as repeating 6's) 4 4 4 4 4 next digit: 0 0 four 8's (one fewer than repeating 6's) 8 8 8 8 a final 96 9 6 3. So the square of 666664 is 444,440,888,896. Squaring special numbers (6's and final 5) Choose a number with repeating 6's and a final 5. . 2. The square is made up of: same number of 4's as repeating 6's S same number of 2's as repeating 6's oa final 25 t Example: h e If the number to be squared is 6665: The square has: sq three 4's (same number as repeating 6's) 4 4 u 4 three 2's (same number as repeating a 6's) 2 2 2 A final 25 2 5 r e of 6665 is 44,422,225. See the pattern? 1. If the number to be squared is 666665: five 4's (same number as repeating 6's) 4 4 4 4 4 five 2's (same number as repeating 6's) 2 2 2 2 2 A final 25 2 5 2. So the square of 666665 is 444,442,222,225. Squaring special numbers (6's and final 7) Choose a number with repeating 6's and a final 7. 2. The square is made up of: The same number of 4's as there are digits in the number; One fewer 8; A final 9. Example: If the number to be squared is 6667: The square has:
- 175. four 4's (numberof digits in number) 4 4 4 4 three 8's (one fewer) 8 8 8 a final 9 3. So the square of 6667 is 44448889. See the pattern? If the number to be squared is 667: The square has: ring special numbers (6's and final 8) Choose a number with repeating 6's and a final 8. 2. The square is made up of: the same number of 4's as there are repeating 6's in the number; � one 6 the same number of 2's as repeating 6's; a final 4. Example: If the number to be squared is 6668: The square has: three 4's (same as repeating 6's) 4 4 4 one 6 6 three 2's (same number as repeating 3's) 2 2 2 a final 4 4 3. So the square of 6668 is 44,462,224. See the pattern? 3 If the number to be squared is 666668: . The square has: S o five 4's (same number as S repeating 6's) 4 4 4 4 4 o 6 one 6 6 6 6 five 2's (same number as 6 repeating 6's) 2 2 2 2 2 6 8 a final 4 4 x 6 66668 = 444,446,222,224. Squaring special numbers (6's and final 9)
- 176. Choose a numberwith repeating 6's and a final 9. 2. The square is made up of: the same number of 4's as there are repeating 6's; a 7 � one fewer 5 than there are repeating 6's; A final 61. Example: If the number to be squared is 6669: The square has: same number of 4's as repeating 6's: 4 4 4a 7 7 one fewer 5 than repeating 6's 5 5a final 61 6 1 3. So the square of 6669 is 44,475,561. See the pattern? If the number to be squared is 666669: The square has: same number of 4's as 6's: 4 4 4 4 4a 7 7 one fewer 5 than repeating 6's 5 5 5 5a final 61 6 1 3. So 666,669 x 666,669 = 44,447,555,561. Use the pattern to amaze your friends with your multiplying abilities. Squaring special numbers (9's and final 1) t h Choose a number with repeating 9's and a final 1. e 2. The square is made up of: � one fewer 9 than there are repeating 9's s ame number of 0's as there are 9's in the square A final 81 Example: If the number to be squared is 9991: The square has: One fewer 9 than the repeating 9's: 9 9 82 8 2 same number of 0's as 9's in the square 0 0 a final 81 8 1 3. So 9991 x 9991 = 99820081. See the pattern? If the number to be squared is 999991: The square has: one fewer 9 than the repeating 9's: 9 9 9 9 82 8 2 same number of 0's as 9's in the square 0 0 0 0 a final 81 8 1 3. So 999991 x 999991 = 999982000081.
- 177. Those big productsought to impress your friends, and they will need a BIG calculator to keep up with you! Squaring special numbers (9's and final 2) Choose a number with repeating 9's and a final 2. 2. The square is made up of: � one fewer 9 than there are repeating 9's the same number of 0's as there are 9's in the square A final 64 Example: If the number to be squared is 9992: The square one fewer 9 than the 9 9 84 8 4 same number of 0's as 9's in the square 0 0 a final 64 6 4 3. So 9992 x 9992 = 99840064. See the pattern? If the number to be squared is 999992: The square has: one fewer 9 than the repeating 9's: 9 9 9 9 84 8 4 same number of 0's as 9's in the square 0 0 0 0 a final 64 6 4 3. So 999992 x 999992 = 999984000064. With a little practice you will be finding these huge products with ease. Squaring special numbers (9's and final 3) S eet Choose a number with repeating 9's and a final 3. h 2. The square is made up of: � one fewer 9 than there are repeating 9's e 86 pthe same number of 0's as there are 9's in the square a A final 49 t t Example: e r If the number to be squared is 9993: n The square has: ?
- 178. one fewer 9than the repeating 9's: 9 9 86 8 6 same number of 0's as I 9's f in the square 0 0 a final 49 4 9 t h 3. So 9993 x 9993 = 99860049. e n umber to be squared is 999993: The square has: one fewer 9 than the repeating 9's: 9 9 9 9 86 8 6 same number of 0's as 9's in the square 0 0 0 0 a final 49 4 9 3. So 999993 x 999993 = 999986000049. Using this pattern you will be able to square these large numbers with ease. Squaring special numbers (9's and final 4) Choose a number with repeating 9's and a final 4. 2. The square is made up of: � one fewer 9 than there are repeating 9's 88 the same number of 0's as there are 9's in the square A final 36 Example: If the number to be squared is 9994: The square has: one fewer 9 than the repeating 9's: 9 9 88 8 8 same number of 0's as 9's in the square 0 0 a final 36 3 6 3. So 9994 x 9994 = 99880036. See the pattern? If the number to be squared is 999994: The square has: o ne fewer 9 than the repeating 9's: 9 9 9 9 88 8 8 same number of 0's as 9's in the square 0 0 0 0 a final 36 3 6 3. So 999994 x 999994 = 999988000036. Squaring special numbers (9's and final 5) Choose a number with repeating 9's and a final 5. 2. The square is made up of: same number of 9's as there are repeating 9's same number of 0's a final 25 Example:
- 179. If the numberto be squared is 9995: The square has: same number of 9's as repeating 9's: 9 9 9 same number of 0's as 9's in the square 0 0 0 a final 25 2 5 3. So 9995 x 9995 = 99900025. See the pattern? 1. If the number to be squared is 999995: same number of 9's as repeating 9's: 9 9 9 9 same number of 0's as 9's in the square 0 0 0 0 a final 25 2 5 3. So 999995 x 999995 = 999990000025. Squaring special numbers (9's and final 6) h o o se a number with repeating 9's and a final 6. 2. The square is made up of: same number of 9's as there are repeating 9's a 2 � one fewer 0 than repeating 9's a final 16 Example: If the number to be squared is 9996: The square has: same number of 9's as repeating 9's: 9 9 9 a22 one fewer 0 than there are 9's in the square 0 0 a final 16 1 6 3. So 9996 x 9996 = 99920016. See the pattern? If the number to be squared is 999996: The square has: same number of 9's as repeating 9's: 9 9 9 9 9 a22 one fewer 0 than there are 9's in the square 0 0 0 0 a final 16 1 6 3. So 999996 x 999996 = 999992000016. Squaring special numbers (9's and final 7) A fi n Choose a number with repeating 9's and a final 7. a
- 180. 2. The squareis made up of: l the same number of 9's as there are repeating 9's 9 4 the same number of 0's as there are 9's in the square E xample: If the number to be squared is 9997: The square has: same number of 9's as there are repeating 9's: 9 9 9 4 4 same number of 0's as 9's in the square 0 0 0 a final 9 9 3. So 9997 x 9997 = 99940009. See the pattern? If the number to be squared is 999997: The square has: same number of 9's as there are repeating 9's: 9 9 9 9 9 4 4 same number of 0's as 9's in the square 0 0 0 0 0 a final 9 9 3. So 999997 x 999997 = 999994000009. Learn the pattern and it's easy! Squaring special numbers (9's and final 8) Choose a number with repeating 9's and a final 8. 2. The square is made up of: the same number of 9's as there are repeating 9's the same number of 0's as there are 9's in the square A final 4 Example: If the number to be squared is 9998: The square has: same number of 9's as there are repeating 9's: 9 9 9 6 6 same number of 0's as 9's 0 a final 4 4 3. So 9998 x 9998 = 99,960,004. See the pattern? If the number to be squared is 999998: The square has: same number of 9's as there are repeating 9's: 9 9 9 9 9 6 6 same number of 0's as 9's in the square 0 0 0 0 0 a final 4 4 3. So 999997 x 999997 = 999,996,000,004.
- 181. Subtract from arepeating 8's number, divide by 9, subtract 10 2 Select a number between 2 and 9 digits made up of 8's. 1 Subtract a number equal to 9 minus the number of digits in the number you have selected. Divide by 9. The answer will have one less digit than the original number in the series 98765432. Subtract 10. Example: If the repeating number selected has 4 digits: 8888: Subtract 5 (9 minus the number of digits) and divide by 9. The answer will be 987 (1 less digit in sequence than the 4 digits in the selected number). Subtract 10: 987 - 10 = 977. So (8888-5) / 9 - 10 = 977. See the pattern? If the repeating number selected has 6 digits: 888888: Subtract 3 (9 minus the number of digits) and divide by 9. The answer will be 98765 (1 less digit in sequence than the 6 digits in the selected number). Subtract 10: 98765 - 10 = 98755. So (888888-3) / 9 - 10 = 98755. Extend this exercise by changing the last number to subtract, producing a multitude of easily answered problems. Subtract from a repeating 1's number, divide by 9, subtract S elect a number between 2 and 10 digits made up of 1's. Subtract a number equal to the number of digits in the number you have selected. Divide by 9. The answer will have one less digit than the original number in the series 123456789. Subtract 21. Example:
- 182. If the repeatingnumber selected has 5 digits:11111: Subtract 5 (same as number of digits) and divide by 9. The answer will be 1234 (4 digits). Subtract 21: 1234 – 21 = 1213. So (11111-5)/9 - 21 = 1213. See the pattern? If the repeating number selected has 8 digits:11111111: Subtract 8 (same as number of digits) and divide by 9 The answer will be 1234567 (7 digits). Subtract 21: 1234567 - 21 = 1234546. So (11111111-8)/9 - 21 = 1234546. E For those who enjoy extensions of these patterns: (answers after the division by 9, step 3) t e 11 digit number, answer is 1234567900 12 digit number, n answer is 12345679011 13 digit number, answer is d 123456790122 14 digit number, answer is 1234567901233 t 15 digit number , answer is 12345679012344 h e and ? e xercises using this basic pattern by changing step 4 to add or subtract other numbers. Subtracting the squares of two numbers Select a 2-digit number and then choose a number two larger or two smaller. Multiply the middle number by 4. Example: If the the first number selected is 31: Choose 29. Multiply the number between them by 4: 4 x 30 = 120 So (31)(31) - (29)(29) = 120. See the pattern? If the the first number selected is 63: Choose 61. Multiply the number between them by 4: 4 x 62 = 248 So 63^2 - 61^2 = 248. One more: If the the first number selected is 73: Choose 75. Multiply the number between them by 4: 4 x 74 = 296 4. So (75x75) - (73x73) = 296. Adding even numbers from two through a selected 2-digit even number
- 183. Divide the evennumber by 2 (or multiply by 1/2). . Multiply this result by the next number. T Example: h e If the 2-digit even number selected is 24: n Divide 24 by 2 (24/2 = 12) or multiply by 1/2 (1/2 x 24 = 12). e xt number is 13; 12 x 13 = 156. Ways to multiply 13 by 12: Square 12, then add 12: 12 x 12 = 144, 144 + 12 = 156. Multiply left to right. 12 x 13 can be done in steps: 12 x (10+3) = (12 x 10) + (12 x 3) = 120 + 36 = 156. So the sum of all the even numbers from two through 24 is 156. See the pattern? If the 2-digit even number selected is 42: Divide 42 by 2 (42/2 = 21 ) or multiply by 1/2 (1/2 x 42 = 21). 3. The next number is 22; 21 x 22 = 462. Ways to multiply 22 by 21: Square 21, then add 21: 21 x 21 = 441, 441 + 21 = 462. Multiply left to right. 21 x 22 can be done in steps: 21 x 22 = (21 x 20) + (21 x 2) = 420 + 42 = 462. So the sum of all the even numbers from two through 42 is 462. Squaring numbers made up of threes Choose a a number made up of threes. 2. The square is made up of: a. one fewer 1 than there are repeating 3's b. zero c. one fewer 8 than there are repeating 3's (same as the 1's in the square) d. nine.
- 184. Example: If the numberto be squared is 3333: The square of the number has: three 1' s (one fewer than digits in number) 1 1 1 _ _ _ _ _ next digit is 0 _ _ _ 0 _ _ _ _ three 8's (same number as 1's) _ _ _ _ 8 8 8 _ a final 9 _ _ _ _ _ _ _ 9 3. So 3333 x 3333 = 11108889. See the pattern? If the number to be squared is 333: The square of the number has: two 1's next digit is 0 two 8's a final 9 . So 333 x 333 = 110889. 11_____ ___0___ ____88_ ______9 Multiplying two 2-digit numbers (same 1st digit) S o Select two 2-digit numbers with the same first digit. 4 2 Multiply their second digits (keep the carry). _ _ _ X x Multiply the sum of the second digits by the first digit, 4 add the carry (keep the carry). _ _ X _ 5 = Multiply the first digits (add the carry). X X _ _ 1 8 9Example: 0 . If the first number is 42, choose 45 as the second number (any 2 -digit number with first digit 4). Multiply the last digits: 2 x 5 =
- 185. 10 (keep carry)_ _ _ 0 Multiply the sum of the 2nd digits by the first: 5 + 2 = 7; 7 x 4 = 28; 28 + 1 = 29 (keep carry) __9_ Multiply the first digits (add the carry) 4 x 4 = 16; 16 + 2 = 18 18__ If See the pattern? t h e first number is 62, choose 67 as the second number (any 2 -digit number with first digit 6). Multiply the last digits: 2 x 7 = 14 (keep carry) _ _ _ 4 Multiply the sum of the 2nd digits by the first (add carry): 2 + 7 = 9; 6 x 9 = 54; 54 + 1 = 55 (keep carry) _ _ 5 _ Multiply the first digits (add the carry) 6 x 6 = 36; 36 + 5 = 41 4 1 _ _ So 62 x 67 = 4154. Multiplying two 2-digit numbers (same 2nd digit) Both numbers should have the same second digit. Choose first digits whose sum is 10. Multiply the first digits and add one second: X X _ _.Multiply the second digits together: _ _ X X. Example: If the first number is 67, choose 47 as the second number (same second digit, first digits add to10). Multiply the 1st digits, add one 2nd. 6x4 = 24, 24+7 = 31. 3 1 _ _ Multiply the 2nd digits. 7x7 = 49 _ _ 4 9 So 67 x 47 = 3149. See the pattern? If the first number is 93, choose 13 as the second number (same second digit, first digits add to10). Multiply the 1st digits, add one 2nd. 9x1 = 9, 9+3 = 12. 1 2 _ _ Multiply the 2nd digits. 3x3 = 9 _ _ 0 9 4. So 93 x 13 = 1209. Multiplying two 2-digit numbers M (same 1st digit, 2nd digits sum to 10) u
- 186. lt Both numbersshould have the same first digit. Choose second digits whose sum is 10. p ly the first digit by one number greater than itself; this number will be the first part of the answer: X X _ _. Multiply the two second digits together; the product will be the last part of the answer: _ _ X X. Note: If the two second digits are 1 and 9 (or, more generally, have a product that is less than ten), insert a 0 (zero) for the first X in step 4. (Thanks to Michael Richardson, age 10, for this note.) Example: If the first number is 47, choose 43 as the second number (same first digit, second digits add to10). 4 x 5 = 20 (multiply the first digit by one number greater than itself): the first part of the answer is 2 0 _ _. 7 x 3 = 21 (multiply the two second digits together); the last part of the answer is _ _ 2 1. So 47 x 43 = 2021. See the pattern? If the first number is 62, choose 68 as the second number (same first digit, second digits add to10). 6 x 7 = 42 (multiply the first digit by one greater), the first part of the answer is 4 2 _ _. 2 x 8 = 16 (multiply the two second digits together); the last part of the answer is _ _ 1 6. So 62 x 68 = 4216. Multiplying two selected 3-digit numbers T (middle digit 0) h e Select a 3-digit number with a middle digit of 0. l Choose a multiplier with the same first two digits, whose third digit sums to 10 with the third a digit of the first 3-digit number. s The first digit(s) will be the square of the first digit: t X _ _ _ _ or X X _ _ _ _. t w The next digit will be the first digit o f the numbers: o _ X _ _ _ or _ _ X _ _ _. d i The next digit is zero: _ _ 0 _ _ or _ _ _ 0 _ _. g
- 187. its will bethe product of the third digits: _ _ _ X X or _ _ _ _ X X.Example: If the first number is 407, choose 403 as the second number (same first digits, second digits add to 10). 4 x 4 = 16 (square the first digit): 1 6 _ _ _ _. The next digit will be the first digit of the numbers: _ _ 4 _ _ . The next digit is zero: _ _ 0 _ _ . 7 x 3 = 21 (the last two digits will be the product of the third digits: _ _ _ 2 1. So 407 x 403 = 164021. See the pattern? If the first number is 201, choose 209 as the second number (same first digits, second digits add to 10). 2 x 2 = 4 (square the first digit): 4 _ _ _ _. The next digit will be the first digit of the numbers: _ 2 _ _ _ . The next digit is zero: _ _ 0 _ _ . 1 x 9 = 09 (the last two digits will be the product of the third digits: _ _ _ 0 9. So 201 x 209 = 42009. Multiplying two selected 3-digit numbers (middle digit 1) Select a 3-digit number with a middle digit of 1. Choose a multiplier with the same first two digits, whose third digit sums to 10 with the third digit of the first 3-digit number. The last two digits will be the product of the first digits: _ _ _ 0 X or _ _ _ _ X X. The third digit from the right will be 2: _ _ 2 _ _ or _ _ _ 2 _ _ . The next digit to the left will be 3 times the first digit of the number (keep carry): _ X _ _ _ or _ _ X _ _ _. The first digits will be the square of the first digit plus the carry: X _ _ _ _ or X X _ _ _ _. As you determine the digits in the answer from right to left, repeat them to yourself at each step until you have the whole answer. Example:
- 188. If the firstnumber is 814, choose 816 as the second number (same first digits, second digits add to 10). 4 x 6 = 24 (multiply the first digits) - last two digits: _ _ _ _ 2 4. The third digit from the right is 2: _ _ 2 _ _ . D K TIWARI CFA ~MODERATOR~ SBI+ALL BANK PO ASPIRANTS COMMUNITY