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Math book

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  • 1. Hi,This book is based on some quicker methodsfor different numerical problems, hence useful for BankExams.MULTIPLICATION TABELE (10x10)1 2 3 4 5 6 7 8 9 101 1 2 3 4 5 6 7 8 9 102 4 6 8 10 12 14 16 18 203 9 12 15 18 21 24 27 304 16 20 24 28 32 36 405 25 30 35 40 45 506 ThinkAs8*44 *8,..36 42 48 54 607 49 56 63 708 68 72 809 81 9010 100MULTIPLICATION TABLE (25x10)21 21 42 63 84 105 126 147 168 189 21022 22 44 66 88 110 132 154 176 198 22023 23 46 69 92 115 138 161 184 207 23024 24 48 72 96 120 144 168 192 216 24025 25 50 75 100 125 150 175 200 225 2501 2 3 4 5 6 7 8 9 1011 11 22 33 44 55 66 77 88 99 11012 12 24 36 48 60 72 84 96 108 12013 13 26 39 52 65 78 91 104 117 13014 14 28 42 56 70 84 98 112 126 14015 15 30 45 60 75 90 105 120 135 15016 16 32 48 64 80 96 112 128 144 16017 17 34 51 68 85 102 119 136 153 17018 18 36 54 72 90 108 126 144 162 18019 19 38 57 76 95 114 133 152 171 19020 20 40 60 80 100 120 140 160 180 200MULTIPLICATION TABLE (11-20 x11-20)11 12 13 14 15 16 17 18 19 20
  • 2. 11 121 132 143 154 165 176 187 198 209 22012 144 156 168 180 192 204 216 228 24013 169 182 195 208 221 234 247 26014 196 210 224 238 252 266 28015 225 240 255 270 285 30016 256 272 288 304 32017 289 306 323 34018 324 342 36019 361 38020 400For quickest math-1. Many diagrams are given in almost every example which would be the reminder of theprocess when ucompletely learn the techniques.2. Try to understand the colors, every color is explaining the steps and other important things.Edited byRAKES PRASADFirst 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 followsNo Square No Sq No Sq No Sqno sq1 1 9 81 17 289 25 625 33 10892 4 10 100 18 324 26 676 34 11563 9 11 121 19 361 27 729 35 12254 16 12 144 20 400 28 784 36 12965 25 13 169 21 441 29 841 38 14446 36 14 196 22 484 30 900 41 16817 49 15 225 23 529 31 9618 64 16 256 24 576 32 1024N.B. The square of 38 contains 444 and the sq of an odd prime no 41 contains two perfectsquares as 42= 16 and 92= 81.Rule No 2. Squares of numbers ending in 5 :we are showing by givinganexample.For the number 25, the last digit is 5 and the previous digit is 2. to multiply the previous digit 2by onemore , 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 resultis
  • 3. 52, that is, 25. Thus 252= 2 X 3 / 25 = 625. In the same way, 1052= 10 X 11/25 = 11025; 1352= 13 X14/25 = 18225; see the figure below…….Now we are extending the formula asRule no 3: If sum of the last two digits give 10 then you can use the formula butL.H.S shouldbe same. Check the examplesEx 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 682 + 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 123127 x 123 = 12 x 13 / 7 x 3 = 156 / 21 = 15621. (see the fig )Example 4. 39523952= 395 x 395 = 39 x 40 / 5 x 5 = 1560 / 25 = 156025.You can rem the formula as 1252=(122+12)25=(144+12)25=15625. Bothare same things but it can use in bigger case easily.Rule 4: Also we can extend the formula where the sum of last two digitsbeing 5 asEx 1. 82 x 83=(82+8/2) /3 x 2=(64+4) / 06=6806 i.e.( n2+n/2) n is even but note thatthe R.H.S term should be of two digits as 06 .( in this ex.)
  • 4. Ex 2.181 x 184=(182+9) / 04=(324+9) / 04=33304.But we know what are u thinking of? Yes, if the number be odd then whathappen?And ur ans is as follows:Ex 3: 91 x 94= (92+9/2) / 04=(81+4) / 54. Surprised? We just pass ½ as 5 in the nextrow since ½ belongs to hundred’s place so 1/2 x 100=50. Just pass 5 in suchcases.See the fig.Ex 4 . 51 x 54= (25+2) / 54 =2754 Ex 5. 171 x 174= (289+8) /54=29754Rule no 5 . Now we are discussing the general method of finding squares ofanynumber. See the chart first.See 52=25And 102= 100 , so on.Now u can understand the tricks.base range Trick alternative50 26-74 25(±)( read as 25 plusminus)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. 432
  • 5. step 1. Find base –no as 50-43=07Step 2. Sq the no as 072=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. 1849Ex 2. 572step 1. 57-50=7Step2. 72=49Step3. 25+7=32step4. Ans is 3249.Ex 3. 692step1. 69-60=19 , 192=361Step2. carry over 3 and write 61 in right side.Step3 . 25+19=44, now add 44 with 3 (carry number) and writeit with extreme left.i.e. (25+19+3)Step 4. ur ans is 4761Ex4 . 38 2 step1. 50-38=12 , 122=144 ,carry 1 and write 44 at right side .Step2. now (25-12)+1=14 i.e. Carry number always adds (itnever 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 morethen addition rule and if it be less, the n subtraction rule will beapplied. That’s why (±) sign is given2. 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 = 12255. for(±) divide the base without zero by 5 as for base 100: 10/5=2,so trickis100 (±)2(±) and for base 350: 35/5=7 ,and the trick is 1225(±)7(±) andso on.
  • 6. Now we are giving more examples to clear the fact, a regular practice can makeumaster of the art.ok, seeThe following examples.For base 100Ex 5. 892Step1. 100-89=11 ,112=121 ,carry 1Step2. 100-(2 x 11 )+1=79, so the ans is 7921Also u can use the alternative formula here,”given no (±)” such asStep 1. 100-89=11,sq is 121,carry 1Step2. 89-11+1=79 ,ass is 7921(I think this process is better for numbers whose base is near 100)Ex 6. 117²Step1. 117-17=17, 172=289 ,carry 2Step2. 117+17=134, 134+2=136 ,ans is 13689For base 150Ex 6. 139²1. 150-139=11 , sq is 121 ,carry 12. 225-3×11=225-33=192, 192+1=193 ,ans is 19, 321Ex 7. 164 ² 1. 164-150=14, 14 ² =196, carry 12. 225+3(14)=225+42=267 ,267+1=268 ,ans is 26,896Ex 8. 512 ² 1. 12 ² =144 ,carry 1, 2500+10(12)+1=2621, ans is 262144Try yourself to various numbers to learn the technique quickly. now we learntofind the square root .before starting note that…Rule no 6:unit digit of perfectsquare1 4569unit digit of square root 1,9(1+9)=102,82+8=1055+5=104,64+6=10
  • 7. 3,73+7=10Notes : 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 range1²=1 1-3 9²=81 81-99 17²=289 289-323 25²=625 625-6752²=4 4-8 10²=100 100-120 18²=324 324-360 26²=676 676-7283²=9 9-15 11²=121 121-143 19²=361 361-399 27²=729 729-7834²=16 16-24 12²=144 144-168 20²=400 400-440 28²=784 784-8405²=25 25-35 13²=169 169-195 21²=441 441-483 29²=841 841-8996²=36 36-48 14²=196 196-224 22²=484 484-528 30²=900 900-9607²=49 49-63 15²=225 225-255 23²=529 529-575 31²=961 961-10238²=64 64-80 16²=256 256-288 24²=576 576-624 32²=1024 1024-1088Rule: 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 separatednumber 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 / 012. 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 51Ex2. √ (6241)1. 62 / 412. 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 rightsideEx3. √ 27041. 27 / 042. 27 falls in range of 5, so 5×6=30 ,but 27 is less than 30, soWe choose 2 between 2 & 8 (allocated for 4)3.so the ans is 52But it is easy if the number ends with 5 such asEx 4. √ (99225)1. 992 /252. Here u can insert 5 as right side because there is only 5 allocatedfor 5 and no choice need here.
  • 8. 3. clearly 992 falls in the range of 31 and the ans is 315Exercise 1. √ 34225 2. √ 105625 3. √ (0.00126025) 4. √ 22095. √ ___________________2916 6. √ 2116 7. √ 15129 8. √ 161299. √ 55696 10. √ 66564 11. √ 8.8804 12. √ 0.0010112413. √ 0.125316Rule no 7: let us know how to find cube root quickly. See the box first.unit digit ofperfect cube1 2 3 4 5 6 7 8 9unit digit ofcube root1 82+8=1073+7=104 5 6 37+3=1028+2=109n.b. 1, 4 , 5, 6, 9 has no change as they were in the sq formula and 2,3,7,8has therecompliment with 10, i.e., 2+8=10, 3+7=10 etc.Ex1. (46656)1/31. 46 / 6562. write 6 for 6 on right side3. see 46 falls in the range of 3 34.so the ans is 36cube range cube range cube range1 =1 1-7 6 =216 216-342 11 =1331 1331-17272 =8 8-26 7 =343 343-511 12 =1728 1728-21963 =27 27-63 8 =512 512-728 13 =2197 2197-27434 =64 64-124 9 =729 729-999 14 =2744 2744-33745 =125 125-215 10 =1000 1000-1330 15 =3375 3375-409521=926125=15625Ex2 . cube of 0.00200483831. 0.0020048 / 3832. write 7 for 3 , and see 2048 falls in the range of 12
  • 9. 3. so the ns be 0 .123n.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) 3Step1. 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 inhorizontal line which are multiplied by the ratio. in this ex asStep 3. Make double of second and third terms and write them just belowtheir own positionsStep 4. Add successive terms (carry over if more than one digit) and u willget the required result .Lets see the methods …1. 18 , that is 1:84 24 512. 1 3 =1 / 1x8=8 / 8 x 8=64 / 64 x 8 =5123. 8 x 2=16 64 x 2=1284. ------------------------------------------------------------( 4 +1) ( 24 +8+16) ( 51 +64+128) 51 25 = 4 8 = 24 35. So the ans is 5832.Ex 2. (33)3Rakes Prasad , 2008It contains some vedic methds for multiplication of two,threeandmore digits,finding h.c.f. ofr two polynomial equations.Edited byRakes Prasad, 2008Rule no (1) If R.H.S. contains less number of digits than the number ofzeros in the base, the remaining digits are filled up by giving zero or zeroeson theleft side of the R.Note: If the number of digits are more than the number of zeroes in hebase, the excess digit or digits are to be added to L.H.S of theanswer.Case 1: Let N1 and N2 be two numbers near to a given base inpowers of 10, and D1 and D2 are their respectivedeviations(difference)from the base. Then N1 X N2 can be represented asEx. 1: Find 97 X 94. Here base is 100.
  • 10. .Ex. 2: 98 X 97 Base is 100.ans: 95/06=9506Ex. 3: 75X95. Base is 100. .Ex. 4: 986 X 989. Base is 1000Ex. 5: 994X988. Base is 1000 .Ex. 6: 750X995Case 2:Ex. 7: 13X12. Base is 10.Ex. 8: 18X14. Base is 10Ex. 9: 104X102. Base is 100.104 04102 02¯¯¯¯¯¯¯¯¯¯¯¯Ans is 106 / 4x2 = 10608 ( rule - f ) .Ex. 10: 1275X1004. Base is 1000.1275 2751004 004...............................................1279 /275x4 =1279 / 1 / 100 (rule f) =1280100Case ( iii ): One number is more and the other is less than the base.Ex.11: 13X7. Base is 10Ex. 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 9944) 1234 X 1002 5) 1003 X 997 6) 11112 X 99987) 1234 X 1002 8) 118 X 105Rules no 2,3,4 of chapter no 1 can also be extended asEg. 1: consider 292 x 208. Here 92 + 08 = 100, L.H.S portionis same i.e. 2292 x 208 = ( 2 x 3 ) / 92 x 8 / =736 ( for 100 raise theL.H.S. product by 0 ) = 60736.Eg. 2: 848 X 852Here 48 + 52 = 100, L.H.S portion is 8 and its next number is 9.848 x 852 = 8 x 9 / 48 x 52720 = 2496= 722496.[Since L.H.S product is to be multiplied by 10 and 2 to be carriedover as the base is 100].Eg. 3: 693 x 607693 x 607 = 6 x 7 / 93 x 7 = 420 / 651 = 420651.Find the following products .
  • 11. 1. 318 x 312 2. 425 x 475 3. 796 x 7444. 902 x 998 5. 397 x 393 6. 551 x 549(2) One less than the previous1) The use of this sutra in case of multiplication by 9,99,999.. is asfollowsa) The left hand side digit (digits) is ( are) obtained by deduction 1fromthe left side digit (digits) .b) The right hand side digit is the complement or difference betweenthemultiplier and the left hand side digit (digits)c) The two numbers give the answerExample 1: 8 x 9Step ( a ) gives 8 – 1 = 7 ( L.H.S. Digit )Step ( b ) gives 9 – 7 = 2 ( R.H.S. Digit )Step ( c ) gives the answer 72Example 2: 15 x 99Step ( a ) : 15 – 1 = 14 Step ( b ) : 99 – 14 = 85 ( or 100 – 15 )Step ( c ) : 15 x 99 = 1485Example 3: 24 x 99 Answer :Example 5: 878 x 9999 Answer :find out the products64 x 99 723 x 999 3251 x 999943 x 999 256 x 9999 1857 x 99999(3) Multiplication of two 2 digit numbers.Ex.1: Find the product 14 X 12The symbols are operated from right to left .Step i) : Step ii) :Step iii) :Ex.4: 32 X 24Step (i) : 2 X 4 = 8Step (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 beadded. i.e., 6 + 1 = 7. Thus 32 X 24 = 768Note 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 andadded.After sufficient practice, you feel no necessity of writing in this wayandsimply operate or perform mentally.(4) Consider the multiplication of two 3 digit numbers.
  • 12. Ex 1. 124 X 132 =16368.Proceeding from right to lefti) 4 X 2 = 8. First digit = 8ii) (2 X 2) + (3 X 4) = 4 + 12 = 16. The digit 6 is retained and 1 is carriedover to left side. Second digit = 6.iii) (1 X 2) + (2 X 3) + (1 X 4) = 2 + 6 + 4 =12. The carried over 1 ofabovestep is added i.e., 12 + 1 = 13. Now 3 is retained and 1 is carried overto left side. Thus third digit = 3.iv) ( 1X 3 ) + ( 2 X 1 ) = 3 + 2 = 5. the carried over 1 of above step isadded i.e., 5 + 1 = 6 . It is retained. Thus fourth digit = 6v) ( 1 X 1 ) = 1. As there is no carried over number from the previousstep 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 thesurplus i.e. 106+12 = 118. (Recall in squaring, we directly addthe 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 multipliedby 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 proceedslike 118 / 108 /....iii) Write down the cube of initial surplus i.e. 63 = 216 as the lastportioni.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 / 216Now 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): 943Observe 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 =108iii) deficit3 = (-6)3 = -216.__ Hence the answer is 82 / 108 / -216Since 100 is base 1 and -2 are the carried over. Adjusting the carriedover
  • 13. in order, we get the answer ( 82 + 1 ) / ( 08 – 03 ) / ( 100 – 16 )= 83 / 05 / 84 = 830584 . 16 becomes 84 after taking1 from middlemost portion i.e. 100. (100-16=84). _ Now 08 - 01 = 07 remains in themiddle portion, and 2 or 2 carried to it makes the middleas 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. 926. 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 ( 1x2 + 5x + 4___________2x + 2 ) x2 + 5x + 4 ( ½xx2 + x__________4x + 4 ) 2x + 2 ( ½2x + 2______0Thus 4x + 4 i.e., ( x + 1 ) is H.C.F.OUR PROCESSi.e.,, (x + 1) is H.C.FExample 2: Find H.C.F. of 2x2 – x – 3 and 2x2 + x – 6Example 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 + 9Add: (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. ÷ - 6 gives 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 + 82 x3 – 3x2 – 4x + 12, x3 – 7x2 + 16x - 123 x3 + 6x2 + 11x + 6, x3 – x2 - 10x - 84 6x4 – 11x3 + 16x2 – 22x + 8, 6x4 – 11x3 – 8x2 + 22x – 8.Edited byRakes Prasad , 2008Contains: (1) Use the formula ALL FROM 9 AND THE LASTFROM 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 SQUAREAND 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 SOFAR:Edited byRakes Prasad, 2008(1) Use the formula ALL FROM 9 AND THE LAST FROM 10 toperforminstant subtractions.● For example 1000 - 357 = 643 We simply take each figure in 357 from 9 and thelast figurefrom 10.So the answer is 1000 - 357 = 643And thats all there is to it!This always works for subtractions from numbers consisting of a 1 followed bynoughts: 100;1000; 10,000 etc.
  • 15. Similarly 10,000 - 1049 = 8951For 1000 - 83, in which we have more zeros than figures in the numbers beingsubtracted, wesimply suppose 83 is 083. So 1000 - 83 becomes 1000 - 083 = 917Try 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 = 10712The answer is in two parts: 107 and 12, 107 is just 103 + 4 (or 104 + 3), and 12 isjust 3 x 4.● Similarly 107 x 106 = 11342 107 + 6 = 113 and 7 x 6 = 42Again, 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 ofthe two figures between the 2 figures.● 26 x 11 = 286Notice that the outerfigures in 286 are the26 being multiplied.And the middle figure is just 2 and 6 added up.● So 72 x 11 = 792Multiply by 11:1) 43 = ) 81 = 3) 15 = 4) 44 = 5) 11 =● 77 x 11 = 847This 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 the 2 and the 4 at the ends. We add the first pair 2 + 3 = 5. and we add thelast 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 5The first figure of 23 is 2, and this is the answer. The remainder is just 2 and 3added up!● 43 / 9 = 4 remainder 7The 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 8The answer consists of 1,4 and 8.1 is just the first figure of 134.4 is the total of thefirst twofigures 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 14Actually a remainder of 9 or more is not usually permitted because we are trying tofind howmany 9s there are in 842.Since the remainder, 14 has one more 9 with 5 left over the final answer will be 93remainder5Divide these by 9:1) 771 2) 942 3) 565 4) 555 5) 777 6) 2382(6) SOME BASIC RELATIONS AND REVIEW OF SQUARE AND CUBEFORMULA:Relations :If a = b , b = c then a= cIf a > b , b > c then a> cIf a < b , b < c then a < cIf a b< b c then a < cIf ab > bc then b > cIf a > b , c> d then a + c> b+ dIf a > b , c< d then a -c> b –dIf a < b , c< d then a -c< b -dIf a > b & a ,b both positive then 1/a < 1/bx = 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 2Factorizationa –b = ( a + b )(a –b) 2 2a + b = (a + b) ( a -ab + b ) 3 3 2 2a -b = (a -b ) (a + ab+ b ) 3 3 2 2Identities(a+ b) + (a –b) = 2(a + b ) 2 2 2 2(a+ b) -(a–b) = 4ab 2 2In dicesamxan=a(m+n)am/an=a(m-n)a=1 0a =1/a -mma = m√a 1/m(a *b ) = amx b m m(a /b ) = a mm/bm(a ) = a m n m*nLogarithmsa = n then log a n= x xlog a (mn )= log am + log anlog a (m/n) = log a m -log anlog a (m) = n log am nlog b n= log an/log abSurd( √a ) = a n nn√ a * √ b = √ab n nn√ a / √ b = √a /b n nm√ √ a = √a = √ √ a n mn n mm√ √ a = √a = √a p p mp p mAngle MeasurementTotal of Interior Angles in DegreesTriangle 180Rectangle 360Square 360Pentagon 540Circle 360Some FactsIn a triangle, interior opposite angle is always less than exterior angle.
  • 18. Sum of 2 interior 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.SquaresNumber Square Number Square Number Square1 1 11 121 21 4412 4 12 144 22 4843 9 13 169 23 5294 16 14 196 24 5765 25 15 225 25 6256 36 16 256 26 6767 49 17 289 27 7298 64 18 324 28 7849 81 19 361 29 84110 100 20 400 30 900Cubes & Other PowersNumber ( X) X3 X4 X511112 8 16 323 27 81 2434 64 256 10245 125 625 31256 216 1296 77767 343 2401 168078 512 4096 327689 729 6561 5904910 1000 10000 100000(7) SQUARE ROOTS AND MULTIPLICATION FORMULA :No Roots No Roots No Roots No Roots1 1 5 2.23 9 3 36 62 1.41 6 2.44 10 3.16 49 73 1.73 7 2.64 16 4 64 84 2 8 2.82 25 5 81 91 to 10 :1 2 3 4 5 6 7 8 9 101 1 2 3 4 5 6 7 8 9 102 2 4 6 8 10 12 14 16 18 203 3 6 9 12 15 18 21 24 27 304 4 8 12 16 20 24 28 32 36 405 5 10 15 20 25 30 35 40 45 506 6 12 18 24 30 36 42 48 54 607 7 14 21 28 35 42 49 56 63 708 8 16 24 32 40 48 56 64 72 809 9 18 27 36 45 54 63 72 81 9010 10 20 30 40 50 60 70 80 90 10011to 20
  • 19. 11 12 13 14 15 16 17 18 19 201 11 12 13 14 15 16 17 18 19 202 22 24 26 28 30 32 34 36 38 403 33 36 39 42 45 48 51 54 57 604 44 48 52 56 60 64 68 72 76 805 55 60 65 70 75 80 85 90 95 1006 66 72 78 84 90 96 102 108 114 1207 77 84 91 98 105 112 119 126 133 1408 88 96 104 112 120 128 136 144 152 1609 99 108 117 126 135 144 153 162 171 18010 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 ENDEdited byRakes Prasad ,2008Contains: 1. Some different techniques of squaringending with 1,2,3,4,6,7,8,9,etc .2. adding different number sequence3. finding the difference of squares.4. dividing a 6-digit/3-digit number by13,7,37037,11,41,15873 etc5. 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 by99,72,84,………9. squaring of some specialnumbersRakes Prasad2008.Squaring a 2-digit number ending in 1
  • 20. .Take a 2-digit number ending in 1. Subtract 1 from thenumber. Squarethe 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 thedifference). 1600 + 40 + 40 = 1680 (add the difference twice toits 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 itssquare). . So 71 x 71= 5041.Squaring a 2-digit number ending in 2I Take a 2-digit number ending in 2. f The last digit will be _ __ 4. tMultiply the first digit by 4: the 2nd number will be h the nextto the lastdigit: _ _ X 4. e n Square the first digit and add the numbercarried from uthe previous step: X X _ _. m b e Example: r is 52, the last digit is _ _ _ 4. 4 x 5 = 20 (four times the firstdigit): _ _ 0 4. 5x 5 = 25 (square the first digit), 25 + 2 = 27 (add carry): 2 7 0 4.For 82 x82, the last digit is _ _ _ 4. 4 x 8 = 32 (four times the firstdigit): _ _ 2 4. 8 x8 = 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-digitnumberending in 3. 2 The last digit will be _ _ _ 9. 3 Multiply the firstdigit by 6:
  • 21. the 2nd number will be the next to the last digit: _ _ X 9. 4Square the firstdigit 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 (sixtimes thefirst digit): _ _ 4 9. 3 4 x 4 = 16 (square the first digit), 16 + 2 =18 (addcarry): 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 timesthe 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. 4So 83 x 83 = 6889.Squaring a 2-digit number ending in 41 Take a 2-digit number ending in 4. 2 Square the 4; the lastdigit is 6: _ __ 6 (keep carry, 1.)3 Multiply the first digit by 8 and add the carry (1); the 2ndnumber will bethe next to the last digit: _ _ X 6 (keep carry). 4 Square thefirst digit andadd the carry: X X _ _.Example:1 If the number is 34, 4 x 4 = 16 (keep carry, 1); the last digitis _ _ _ 6. 2 8x 3 = 24 (multiply the first digit by 8), 24 + 1 = 25 (add thecarry): the nextdigit is 5: _ _ 5 6. (Keep carry, 2.) 3 Square the first digit andadd the
  • 22. carry, 2: 1 1 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 seconddigit (keepthe carry): the last digit of the answer is always 6: _ _ _ 63 Multiply the first digit by 2 and add the carry (keep thecarry): _ _ X _4 Multiply the first digit by the next consecutive number andadd thecarry: the product is the first two digits: XX _ _.Example:1 If the number is 46, square the second digit : 6 x 6 = 36; thelast digit ofthe answer is 6 (keep carry 3): _ _ _ 6 2 Multiply the first digit(4) by 2 andadd the carry (keep the carry): 2 x 4 = 8, 8 + 3 = 11; the nextdigit of theanswer is 1: _ _ 1 63 Multiply the first digit (4) by the next number (5) and addthe carry: 4 x5 = 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; thelast digit ofthe answer is 6: _ _ _ 6 2 Multiply the first digit (7) by 2 andadd the carry:
  • 23. 2 x 7 = 14, 14 + 3 = 17; the next digit of the answer is 7 (keepcarry 1): _ _7 6 3 Multiply the first digit (7) by the next number (8) and addthe 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 theanswer isalways 9: _ _ _ 9 3 Multiply the first digit by 4 and add 4 (keepthe carry):__X_4 Multiply the first digit by the next consecutive number andadd thecarry: 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 Multiplythe 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 94 Multiply the first digit (4) by the next number (5) and addthe carry (2): 4x 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 3Multiply the firstdigit (6) by 4 and add 4 (keep the carry): 4 x 6 = 24, 24 + 4 =28; the nextdigit of the answer is 0 (keep carry 2): _ _ 8 9 4 Multiply thefirst digit (6)by the next number (7) and add5 the carry (2): 6 x 7 = 42, 42 + 2 = 44 (the first two digits): 4 6So 67 x 67= 4489.
  • 24. 1 Choose a 2-digit number ending in 8. 2 The last digit of theanswer isalways 4: _ _ _ 4 3 Multiply the first digit by 6 and add 6 (keepthe carry):__X_4 Multiply the first digit by the next consecutive number andadd thecarry: 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 Multiplythe 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 4Multiply the firstdigit (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 38The last digit of the answer is 4: _ _ _ 4Multiply the first digit (3) by 6 and add 6 (keep thecarry): 3 x 6 = 18, 18 + 6 = 24; the next digit of theanswer is 4 (keep carry 2): _ _ 4 4Multiply 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 = 1444Squaring a 2-digit number ending in 91 Choose a 2-digit number ending in 9. 2 The last digit of theansweris always1. Multiply the first digit by 8 and add 8 (keep the carry): _ _ X_3
  • 25. Multiply the first digit by the next consecutive number andadd thecarry: the product is the first two digits: XX _ _.Example:1 If the number is 39: 2 The last digit of the answer is 1: _ _ _13Multiply the first digit (3) by 8 and add 8 (keep the carry): 8 x3=24, 24 + 8 = 32; the next digit of the answer is 2 (keep carry3): _ _2 1 4 Multiply the first digit (3) by the next number (4) and addthecarry (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 792 The last digit of the answer is 1: _ _ _ 1Multiply the first digit (7) by 8 and add 8 (keep the carry): 8 x7 = 56,56 + 8 = 64; the next digit of the answer is 4 (keep carry 6): __41Multiply the first digit (7) by the next number (8) and add thecarry (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 forexperts). Add allthe numbers between them:1 Add the numbers; 2 Subtract the numbers and add 1; 3Multiply halfthe sum by this difference + 1, OR Multiply the sum by halfthe difference+ 1.Example:
  • 26. 1 If the two numbers selected are 6 and 19: 2 Add thenumbers: 6 + 19 =25. 3 Subtract the numbers: 19 - 6 = 13. Add 1: 13 + 1 = 14. 4Multiply 25by half of 14: 25 x 7 = 175. 5 So the sum of the numbers from6 through 19is 175.Therefore 6+7+8+9+10+11+12+13+14+15+16+17+18+19=175See the pattern?1 If the two numbers selected are 4 and 18: 2 Add thenumbers: 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 thesum of thenumbers from 4 through 18 is 165.1 Choose a 2-digit odd number. Add all the odd numbersstarting withone through this 2-digit number: 2 Add one to the 2-digitnumber. 3Divide this sum by 2 (take half of it). 4 Square this number.This is thesum of all odd numbers from 1 through the 2-digit numberchosen.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. 5So the sum of all the odd numbers from one through 35 is324.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(square40).5. So the sum of all the odd numbers from one through 79 is1600.1 Have a friend choose a a single digit number. (Norestrictions forexperts.) 2 Ask your friend to jot down a series of doubles(where thenext term is always double the preceding one), and tell youthe last term.3 Ask your friend to add up all these terms. 4 You will givethe answerbefore he or she can finish: The sum of all the terms of thisseries will betwo 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? Heres 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 x512 = 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. Withpractice you will beexpert in summing series.restrictions for experts.) Ask your friend to jot down a seriesof
  • 28. quadruples (where the next term is always four times thepreceding one),and tell you only the last term. Ask your friend to add up allthese terms.You will give the answer before he or she can finish: Thesum of all theterms of this series will be four times the last term minus thefirst term,divided by 3. Example:If the number selected is 5:1 The series jotted down is: 5, 20, 80, 320, 1280. 2 Four timesthe last term(1280) minus the first (5): 4000 + 800 + 320 - 5 = 5120 - 5 =5115 Divide by3: 5115/3 = 17053 So the susum of the quadruples from 5 through 1280 is1705.See the pattern? Heres 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 = 8,160Divide by 3: 8160/3 = 2720.4. So the sum of the quadruples from 32 through 2048 is2720.Practice multiplying from left to right and dividing by 3. Withpractice youwill be anexpert quad adder.Add a sequence from one to a selected 2-digit number1 Choose a 2-digit number. 2 Multiply the 2-digit number byhalf the nextnumber, orMultiply half the 2-digit number by the nextnumber.
  • 29. Example:1 If the 2-digit even number selected is 51: 2 The nextnumber is 52.Multiply 51 times half of 52. 3 51 x 26: (50 x 20) + (50 x 6) + 1 x26) = 1000+ 300 + 26 = 13264 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 nextnumber 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 = 5954 So the sum of all numbers from 1 through 34 is 595.With some multiplication practice you will be able to findthese sums ofsequential numbers easily and faster than someone using acalculator!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 numbersfrom 1through 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 ofall numbersfrom 1 through 9 and back is 81.Add sequences of numbers in the 10s1 Choose a 2-digit number in the 10s. To add all the 10sfrom 10 up
  • 30. through this number and down from it: 2 Square the 2nd digitof thenumber (keep the carry) _ _ X 3 The number of terms is 2 xthe 2nd digit +1. 4 The first digits = the number of terms (+ carry). nbsp; X X_Example:1 If the 2-digit number in the 10s 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 = 134 No. of terms (+ carry): 13 + 3 = 16 1 6 _ 5 So the sum of thesequence is166.See the pattern?1. If the 2-digit number in the 10s selected is 18:(10 + 11 + 12 + ... 18 + 17 + 16 + 15 ... 10)1 Square the 2nd digit of the number: 8 x 8 = 64 (keep carry6) _ _ 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 20s. To add all the 20sfrom 20 upthrough this number and down from it: 2 Square the 2nd digitof thenumber (keep the carry) _ _ X 3 The number of terms is 2 xthe 2nd digit +1. 4 Multiply the number of terms by 2 (+ carry). X X _Example:1 If the 2-digit number in the 20s selected is 23: (20 + 21 + 22+ 23 + 22 +21 + 20) 2 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 3No. of
  • 31. terms = 2 x 2nd digit + 1: 2 x 3 + 1 = 7 4 2 x no. of terms: 2 x 7= 14 1 4 _ 5So the sum of the sequence is 149.See the pattern?1 If the 2-digit number in the 20s selected is 28: (20 + 21 +22 ... + 28 + 27+ ... 22 + 21 + 20) 2 Square the 2nd digit of the number: 8 x 8= 64 (keepcarry 6) _ _ 4 3 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 174 2 x no. ofterms (+ carry): 2 x 17 + 6 = 40 4 0 _ 5 So the sum of thesequence is 404.2 S 1 Choose a 2-digit number in the 30s. To add all the 30sqfrom 30 upthrough this number and down from it: uare the 2nd digit of the number (keep the carry) _ _ X 3 Thenumber ofterms is 2 x the 2nd digit + 1. 4 Multiply the number of termsby 3 (+carry) X X _Example:1 If the 2-digit number in the 30s selected is 34: (30 + 31 + 32+ 33 + 34 +33 + 32 + 31 + 30) 2 Square the 2nd digit of the number: 4 x 4= 16 (keepcarry 1) _ _ 6 3 No. of terms = 2 x 2nd digit + 1: 2 x 4 + 1 = 94 3 x no. of terms: 3 x 9 + 1 = 28 2 8 _ 5 So the sum of thesequence is286.See the pattern?1 If the 2-digit number in the 30s selected is 38: (30 + 31 + 232 + ... + 38 +37 + ... 32 + 31 + 30) 3 Square the 2nd digit of the number: 8 x8 = 64 (keepcarry 6) _ _ 4
  • 32. No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 173 x no. of terms: 3 x 17 + 6 = 51 + 6 = 57 5 7 _ So the sum ofthe sequenceis 574.1 Choose a 2-digit number in the 40s. To add all the 40sfrom 40 up through this number and down from it:2 Square the 2nd digit of the number(keep the carry) _ _X3 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 40s 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 = 2x 2nd digit + 1: 2 x 3 + 1 = 73 4 x no. of terms: 4 x 7 = 28 2 8 _ 4 So the sum of thesequence is 289.See the pattern?1 If the 2-digit number in the 40s selected is 48: (40 + 41 + 42+ ... + 48 +47 + ... 42 + 41 + 40) 2 Square the 2nd digit of the number: 8 x8 = 64 (keepcarry 6) _ _ 4 3 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 174 4 x no. ofterms: 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 50s. To add all the 50sfrom 50 upthrough this number and down from it: 2 Square the 2nd digitof thenumber (keep the carry) _ _ X 3 The number of terms is 2 xthe 2nd digit +1. 4 Multiply the number of terms by 5. X X _Example:
  • 33. 1 If the 2-digit number in the 50s selected is 53: (50 + 51 + 52+ 53 + 52 +51 + 50) 2 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 3No. ofterms = 2 x 2nd digit + 1: 2 x 3 + 1 = 7 4 5 x no. of terms: 5 x 7= 35 3 5 _ 5So the sum of the sequence is 359.See the pattern?1 If the 2-digit number in the 50s selected is 57: (50 + 51 + 52 ... + 57 + 56 + ... 52 + 51 + 50) 2 Square the 2nd digit of thenumber: 7 x 7= 49 (keep carry 4) _ _ 93 No. of terms = 2 x 2nd digit + 1: 2 x 7 + 1 = 155 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 60s1 Choose a 2-digit number in the 60s. To add all the 60sfrom 60 up throughthis number and down from it: 2 Square the 2nd digit of thenumber (keepthe carry) _ _ X 3 The number of terms is 2 x the 2nd digit + 1.4 Multiply thenumber of terms by 6 (+ carry) X X _Example:1 If the 2-digit number in the 60s 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 x2nd digit + 1: 2 x 3 + 1 = 745 6 x no. of terms: 6 x 7 = 42 4 2 _ So the sum of thesequence is 429.See the pattern?1 If the 2-digit number in the 60s selected is 68: (60 + 61 +62 + ... + 68 + 67 + ... 62 + 61 + 60) 2 Square the 2nd digit of
  • 34. 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 6 x no. of terms: 6 x 17 +6 = 42 + 1 = 102 + 6 = 108 1 0 8 _5oSthe sum of the sequence is 1084.1 Choose a 2-digit number in the 70s. To add all the 70sfrom 70 upthrough this number and down from it: 2 Square the 2nd digitof thenumber (keep the carry) _ _ X3 The number of terms is 2 x the 2nd digit + 1. 4 Multiply thenumber ofterms by 7 (+ carry) X X _Example:1 If the 2-digit number in the 70s selected is 73: (70 + 71 + 72+ 73 + 72 +71 + 70) 2 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 3No. ofterms = 2 x 2nd digit + 1: 2 x 3 + 1 = 74 7 x no. of terms: 7 x 7 = 49 4 9 _ 5 So the sum of thesequence is 499.See the pattern?1 If the 2-digit number in the 70s selected is 78: (70 + 71 + 72+ ... + 78 +77 + ... 72 + 71 + 70) 2 Square the 2nd digit of the number: 8 x8 = 64 (keepcarry 6) _ _ 4 3 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1 = 174 7 x no. ofterms: 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 80s. To add all the80sfrom 80 up through this number and down from it:2 Square the 2nd digit of the number(keep the carry) _ _
  • 35. umber of terms is 2 x the 2nd digit + 1 4 Multiply the numberof terms by8 (add the carry) X X _Example:1 If the 2-digit number in the 80s selected is 83: (80 + 81 + 82+ 83 + 82 +81 + 80) 2 Square the 2nd digit of the number: 3 x 3 = 9 _ _ 9 3No. ofterms = 2 x 2nd digit + 1: 2 x 3 + 1 = 7 4 8 x no. of terms: 8 x 7= 56 5 6 _ 5So the sum of the sequence is 569.See the pattern?1 If the 2-digit number in the 80s selected is 88: (80 + 81 +82 ... + 88 + 87+ ... 83 + 82 + 81 + 80) 2 Square the 2nd digit of the number: 8x 8 = 64(keep carry 6) _ _ 4 3 No. of terms = 2 x 2nd digit + 1: 2 x 8 + 1= 178 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.(Twodigits for experts.) 2 Ask your friend to name and note a thirdnumberby adding the first two. 3 Name a fourth by adding thesecond and third.Continue in this way, announcing each number, through tennumbers. 4Ask your friend to add up the ten numbers. You will give theanswerbefore he or she can finish:The sum of all the terms of this series will be the seventhnumbermultiplied by 11.
  • 36. Example:1 If the numbers selected are 7 and 4:e series jotted down is: 4, 7, 11, 18, 29, 47, 76, 123, 199, 322. 3Theseventh number is 76. 11 x 76 = 836 (use the shortcut for 11:7 is the firstdigit, 6 is the third digit; the middle digit will be 7 + 6, andcarry the 1:836).4 So the sum of the ten numbers is 836.Here are some of the calculations that you can do mentallyafterpracticing the exercises. See how many you can do. Checkyour answerswith a calculator. Your mental math powers should beimpressive!15 x 15 89 x 89 394 x 101 32 x 38 51 x 59101 x101 x 94 93 x 93 228 x 101 25 x 25147 56 x 56 94 x 96 101 x 448 83 x 87 687 x 101 64 x 66 52 x 52101 x 65435 x 35 11 x 19 101 x61 x 69 89 x 101 61 x 61 34 x 36206 101 x 48 52 x 52 45 x 45 456 x 101 54 x 54 41 x 41 101 x 4132 x 32 83 x101 29 x 29 55 x 55 63 x 67 479 x 101 82 x 82 882 x 101 14 x 16319 x 10173 x 77 65 x 65 101 x 859 13 x 17 101 x 149 41 x 49 82 x 101 75x 75 101 x71 x 79 69 x 101 22 x 22 993 x 101144 51 x 51 85 x 85 101 x 101 129 x 101 69 x 69 31 x 39 738 x101 21 x 2971 x 71 77 x 101 95 x 95 265 x 101 74 x 76 53 x 53 101 x 409 51x 51 98 x101 94 x 94 42 x 42 53 x 57 339 x
  • 37. 62 x 68 101 x 88 19 x 19 238 x 10110143 x 47 96 x 96 21 x 21 101 x 279 81 x 89648 x62 x 62 101 x 57 24 x 26 101 x 668101 99 x 99 23 x 27 22 x 28 515 x 101 97 x 97 79 x 79 101 x 826245 x 10131 x 31 37 x 33 101 x82 x 88 39 x 39 126 x 101 91 x 9921868 x 101 51 x 101 598 x 101 98 x 98 72 x 7291 x 91 54 x 56 101 x 29 57 x 57 771 x 101 81 x 81 559 x 101 52x 58 59 x 5984 x 86 46 x 44 101 x 25 12 x 18 101 x 697 92 x 92 349 x93 x 97 101 x 188 42 x 48 49 x 4910158 x 58 92 x 92 101 x 78 37 x 1011 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 yourereally sharp, doit mentally: 24 x 24 = 576, 25 x 25 = 625, 625 - 576 = 49.) 2 If 63and 64 areselected, then 63 + 64 = 127. (For larger number addition, doit in steps:63 + 64 = 63 + 60 + 4 = 123 + 4 = 127.)1 Select two consecutive 2-digit numbers, one not more than10 largerthan the other (experts need not use this limitation). 2Subtract thesmaller number from the larger. 3 Add the two numbers. 4Multiply thefirst answer by the second.Examples:
  • 38. 1 If 71 and 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: 7x 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 TricksIn reading a popular math book, I came across severalarithmeticfactoring tricks. Essentially, if the last digit of a number iszero,then the entire number is divisible by 10. If the last number iseven, then the entire number is divisible by 2. If the last twodigits are divisible by 4, then the whole number is. If the lastthree digits divide by 8, then the whole number does. If thelastfour digits divide by 16, then the whole number does, etc. Ifthelast 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 by3,then the whole number is. Similarly, fif you add the digits ina number and the sum is divisible by 9, then the wholenumber is. For example, take the number 1233: 1 + 2 + 3 + 3 =9.
  • 39. Therefore, the whole number is divisible by 9 and thequotient is137. The number is also divisible by 3 and the quotient is411.It works for extremely large numbers too (I checked on mycalculator). Now heres a really tricky trick. You add up thealternate digits of a number and then add up the other setof alternate digits. If the sums of the alternate digits equaleach other then the whole number is divisible by 11. Also,if the difference of the alternate digits is 11 or a multiple of11, 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 thequotientis 11,223,302. Also, if a number is divisible by both 3 and 2,thenthe whole number is divisible by 6.The only single digit number for which there is no trick listedis 7.I find these rules interesting and useful, especially whenfactoringlarge numbers in algebraic expressions. However, Im notsurewhy all these rules work. Can you explain to me why thesemathtricks work? If I could understand why they work, I think itwouldimprove my math skills. Thanks in advance for your help.Cindy Smith cms@dragon.comFrom: Dr. Math <dr.math@forum.swarthmore.edu> To:cms@dragon.com (Cindy Smith) Subject: Re: Factoring TricksThank you for this long and very well-written question. I willtry towrite as clearly as you as I answer. All these digital tests for
  • 40. divisibility are based on the fact that our system of numeralsiswritten using the base of 10.The digits in a string of digits making up a numeral areactually thecoefficients of a polynomial with 10 substituted for thevariable. Forexample, 1233 = 1*10^3 + 2*10^2 + 3*10^1 + 3*10^0 which isgottenfrom 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 thesetricksin terms of that fact: 10 - numerals ending in 0 representnumbersdivisible by 10:Since the last digit is zero, and all other terms in thepolynomial formare divisible by 10, the number is divisible by 10. Similarly, ifthenumber is divisible by 10, since all the terms except the lastoneare automatically divisible by 10 no matter what thecoefficients ordigits are, the number will be divisible by 10 only if the lastdigit is.Since all the digits are smaller than 10, the last digit has to be0 tobe a multiple of 10.2 - numerals ending in an even digitrepresentnumbers divisible by 2: f o Same argument as above about alltheterms except the last one being divisible by 2. The last digitis
  • 41. divisible by 2 (even) if and only if the whole number is. R 4 - numerals endingwitha two-digit multiple of 4 represent numbers divisible by 4:o fSimilar to the above, but since 10 is not a multiple of 4, but10^2 is,we have to look at the last two digits instead of just the lastdigit.a n 8 - numerals ending with a three-digit multiple of 8representnumbers divisible by 8: u Similar to 4, but now 10^2 is not amultipleof 8, but 10^3 is, so we have to look at the last three mdigits.era16 - You figure this one!l5 - You figure this one, too!ook3 - numerals whose sum of digits is divisible by 3 representnumberss divisible by 3: This one is different, because 3 does notdivide anypower of 10 evenly. That means we will have to consider theeffect ofall 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 thepolynomial 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 firstpart, so thewhole 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 representnumbersdivisible by 9: Use the same equation as the previous case.You figurethe rest!11 - numerals whose alternating sum of digits is divisible by11 representnumbers divisible by 11:Here the phrase "alternating sum" means we alternate thesigns frompositive to negative to positive to negative, and so on. Weuse this fact:10 to an odd power plus 1 is divisible by 11, and 10 to aneven powerminus 1 is divisible by 11. The first part is a fancy way ofwriting 10000...0001 = 11*9090...9091 (where there are an even number of0s on theleft-hand side). The second part is a fancy way of writing99999...9999= 11*9090...0909 (where there are an even number of 9s inthe lefthand 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 substitutethatinto 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 thewhole number is divisible by 11 if and only if the last part,which isthe alternating sum of the digits, isdivisible by 11. If you prefer, you can writed[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 fromthe units digit, and then add up the remaining digits, andsubtract thetwo sums. This will compute the same result as thealternating sum ofthe digits. 7 - There is a trick for 7 which is not as well knownas theothers. It makes use of the fact that 10^(6*k) - 1 is divisible by7, and10^(6*k - 3) + 1 is divisible by 7. It goes like this: Mark off thedigits ingroups of threes, just as you do when you put commas inlarge numbers.Starting from the right, compute the alternating sum of thegroups asthree-digit numbers. If the result is negative, ignore the sign.If theresult is greater than 1000, do the same thing to the resultingnumberuntil you have a result between 0 and 1000 inclusive. That 3-digit
  • 44. number is divisible by 7 if and only if the original number istoo.Example:123471023473 = 123,471,023,473, so make the sum473 - 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 a1, andevery 9 by a 2, before, during, or after the sum, and the factremains.The sum could also have been computed as473 - 23 + 471 - 123 --> 403 - 23 + 401 - 123 = 380 + 278 --> 310+ 201 =511 = 7*73. You can figure out why this "casting out 7s" partworks.There is another way of testing for 7 which uses the fact that7 divides2*10 + 1 = 21. Start with the numeral for the number you wantto test.Chop off the last digit, double it, and subtract that from therest of thenumber. Continue this until you get stuck. The result is 7, 0,or -7, ifand 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 - The same trick that works for 7 works for 13; that is, 13divides10^(6*k) - 1 and 10^(6*k - 3) + 1, so the alternating sum ofthree-digitgroups works here, too. 17 - This is harder. You would havetouse alternating sums of 8-digit groups!1 Select a 3-digit number. 2 Repeat these digits to make a 6-digitnumber. 3 Divide these 6 digits by 7, then by 13. 4 Theanswer is 11 times the first three digits!Example:1 If the 3-digit number selected is 234: 2 The 6-digit numberis 234234.3. Divide by 7, then by 13: multiply by 11-to multiply 234 by 11, work right to left: last digit on right = ___4next 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 numberis 461461.3. Divide by 7, then by 13: multiply by 11-to multiply 461 by 11, work right to left: last digit on right = ___1nextdigit 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 formultiplying anynumber by 11.
  • 46. 1 Select a 3-digit number. 2 Repeat these digits to make a 6-digitnumber. 3 Divide these 6 digits by 13, then by 11. 4 Theansweris 7 times the first three digits!Example:1 If the 3-digit number selected is 231: 2 The 6-digit numberis 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 numberis 412412.Remembertomultiply left to right and add in increments. Then you willbe able to give these answers quickly and accurately.1 Select a 6-digit number repeating number. 2 Repeat thesedigits tomake a 6-digit number. 3 Multiply a single digit by 3, then by5.Example:1 If the 6-digit repeating number selected is 333333: 2Multiply 3 x 3: 9 3Multiply 9 x 5: 45 4 So 333333 divided by 37037 andmultiplied by 5 is 45.See the pattern?You can expand this exercise by using a different number inthe final step.Example: multiply by 4:1 If the 6-digit repeating number selected is 555555: 2Multiply 3 x 5: 15. 3
  • 47. Multiply 15 x 4: 60 4 So 555555 divided by 37037 andmultiplied by 4 is 60.By changing the last step you can generate many extensionsof this3 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 impressivecalculations.Dividing a repeating 6-digit number by 7, 11, 13; subtract 101Select 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 = 188So 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 = 882So 983983 divided by 7, then 11, then 13 minus 101 is 882.1 Select a repeating 3-digit number . 2 The answer is 3 timesoneof 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. 4 So (999 / 37) + 41 = 68.Change the last step to add other numbers, and thusproduce manynew exercises.1 Select a repeating 6-digit number . 2 The answer is 7 timesthe firstdigit of the number!Example:1 777777 / 15873 = 7 x 7 = 49. 2 555555 / 15873 = 7 x 5 = 35. 3999999 /15873 = 7 x 9 = 63. Not very demanding mental math, butgood for aquick challenge or two.Dividing mixed numbers by 21 Select a mixed number (a whole number and a fraction).� 2. If the whole number is even, divide by 2 - this is thewhole numberof the answer.�� The numerator of the fraction stays the same; multiplythedenominator by 2.� 3. If the whole number is odd, subtract 1 and divide by 2 -thisis the whole number of the answer.� Add the numerator and the denominator of the fraction -thiswill 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 Multiplythedenominator by 2: 4 x 2 = 8 (denominator) So 8 3/4 divided by2 = 4 3/8.Odd whole number example:1 If the first number selected is 13 2/5: 2 3 Subtract 1 from thewholenumber and divide by 2: 13 - 1 = 12, 12/2 = 6.Add the numerator and the denominator: 2 + 5 = 7. This is thenumerator of the fraction. Multiply the denominator by 2: 5 x2 = 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 threetimes).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. 2Multiplyby 2 three times: 2 x 78 = 156 2 x 156 = 312 2 x 312 = 6243 Move the decimal point 2 places to the left: 6.24 4 So 78dividedby 12 1/2 = 6.24.See the pattern?1 If the 2-digit number chosen to multiply by 12 1/2 is 91: 2Double three times: 182, 364, 728. 3 Move the decimal point2 places to the left: 7.28 4 So 91 divided by 12 1/2 = 7.28.5ividing a 2-digit number by 151 Select a 2-digit number. 2 Multiply it by 2. 3 Divide theresult 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. 2Multiply by 2:
  • 50. 2 x 68 = 120 + 16 = 136 3 Divide the result by 3: 136/3 = 45 1/34Move the decimal point 1 place to the left: 4.5 1/3 5 So 68divided by 15 = 4.51/3.See the pattern?1 The 2-digit number chosen to multiply by 15 is 96. 2Multiplyby 2: 2 x 96 = 180 + 12 = 192 3 Divide the result by 3: 192/3 =644 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 15with two quick1 Select a 2-digit number. (Choose larger numbers when youfeel sureabout the method.) 2 Multiply by 4 (or by 2 twice). 3 Move thedecimall point two places to the left.Example:1 The 2-digit number chosen to divide by 25 is 38. 2 Multiplyby 4: 4 x 38= 4 x 30 = 120 + 32 = 152. 3 Move the decimal point 2 placesto 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 Multiplyby 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 641divided by25 = 25.64.1 Select a 2-digit number (progress to larger ones). 2 Multiplyit by 3. 3Move the decimal point 2 places to the left.
  • 51. Example:1 The 2-digit number chosen to multiply by 33 1/3 is 46. 2Multiply by 3:3 x 46 = 3(40 + 6) = 120 + 18 = 138 3 Move the decimal point 2places tothe left: 1.38 4 So 46 divided by 33 1/3 = 1.38. (If you divide by33.3using 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: 2Multiplyby 3: 3 x (600 + 50) = 1800 + 150 = 1950 3 Move the decimalpoint 2places to the left: 19.50 4 So 650 divided by 33 1/3 = 19.5.Practice multiplying left to right and this procedure willbecome aneasy one - and you will get exact answers, too.Dividing a 2- or 3-digit number by 351 Select a 2-digit number. (Choose larger numbers when youfeel sureabout the method.) 2 Multiply by 2. 3 Divide the resultingnumber 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 x61 = 122.Divide by 7: 122/7 = 17 3/7 Move the decimal point 1 place tothe 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 1place to theleft: 1.2 4/7 5 So 44 divided by 35 = 1.2 4/7.
  • 52. Division done by calculator will give repeating decimals(unless theoriginal number is a multiple of 7), truncated by the limits ofthe display.The exact answer must be expressed as a mixed number.1 Select a 2-digit number. 2 Multiply by 8. 3 Divide theproduct by 3. 4Example:1 The 2-digit number chosen to divide by 37 1/2 is 32. 2Multiply by 8:8 x 32 = 240 + 16 = 256 3 Divide by 3: 256/3 = 85 1/3 4 Movethe decimalpoint 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. 2Multiply by 8:8 x 51 = 408 3 Divide by 3: 408/3 = 136 4 Move the decimalpoint twoplaces to the left: 1.36 5 So 51 divided by 37 1/2 = 1.36.Dividing a 2-digit number by 451 Select a 2-digit number. 2 Divide by 5. 3 Divide the resultingnumberby 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 32dividedby 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 dividedby 451.35 5/9.
  • 53. 1 Select a 2-digit number. (Choose larger numbers when youfeelsure about the method.) 2 Multiply by 4 (or by 2 twice). 3Move thedecimal point two places to the left. 4 Divide by 3 (expressremainderas a fraction).Example:1 The 2-digit number chosen to divide by 75 is 82. 2 Multiplyby 4:4 x 82 = 328. 3 Move the decimal point 2 places to the left:3.28 4Divide 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 Multiplyby 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 Divideby 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-digit number chosen to divide by 1 1/3 is 47. 2Multiply by 3:3 x 47 = 120 + 21 = 141 3 Divide by 4: 141/4 = 35 1/4 4 So 47dividedby 1 1/3 = 35 1/4.See the pattern?1 The 2-digit number chosen to divide by 1 1/3 is 82. 2Multiply by 3:3 x 82 = 246 3 Divide by 4: 246/4 = 61 1/2 4 So 82 divided by1 1/3 = 61 1/2.With this pattern you will be able to give answers quickly, butmostimportantly, your answers will be exact. If a calculator userdividesby 1.3, the answer will NOT be correct.1 Select a 2-digit number. 2 Multiply by 8. 3 Move the decimalpointone place to the left.Example:1 The 2-digit number chosen to divide by 1 1/4 is 32. 2Multiply by 8:8 x 32 = 240 + 16 = 256 3 Move the decimal point one place tothe 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. 2Multiply by 8:8 x 64 = 480 + 32 = 512 3 Move the decimal point one place tothe left:51.2 4 So 64 divided by 1 1/4 = 51.2.Multiply from left to right for ease and accuracy. You willsoon bedoing 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 number chosen 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 dividedby 1 1/5= 63 1/3.Dividing a 2-digit number by 1 2/31 Select a 2-digit number. 2 Multiply by 6 (or by 2 and 3). 3Move the decimal point one place to the left.Example:1 The 2-digit number chosen to divide by 1 2/3 is 78. 2Multiplyby 3: 3 x 78 = 210 + 24 = 234 3 Multiply by 2: 2 x 234 = 468 4Movethe decimal point one place to the left: 46.8 5 So 78 dividedby1 2/3 = 46.8.See the pattern?1 The 2-digit number chosen to divide by 1 2/3 is 32. 2Multiply by 3:3 x 32 = 96 3 Multiply by 2: 2 x 96 = 180 + 12 = 192 4 Move thedecimalpoint 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 becomeadept atmentally dividing a number by 1 2/3.1 Select a 2-digit number. 2 Multiply by 8. 3 Move the decimalpoint3 places to the left.Example:
  • 56. 1 The 2-digit number chosen to divide by 125 is 72. 2 Multiplyby 8:8 x 72 = 560 + 16 = 576. 3 Move the decimal point 3 places tothe 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 Multiplyby 8:8 x 42 = 320 + 16 = 336.3 ove the decimal point 3 places to the left: .336 4 So 42divided by125 = .336.1 Select a 2-digit number. 2 Multiply by 4. 3 Divide theproduct by 7.Example:1 The 2-digit number chosen to divide by 1 3/4 is 34. 2Multiply by 4:4 x 34 = 136 3 Divide the product by 7: 137/6 = 19 3/7 (If youuse acalculator, 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. 2Multiply by 4:4 x 56 = 224 3 Divide the product by 7: 224/7 = 32 4 So 56dividedby 1 3/4 = 32.Notice that numbers divisible by 7 will produce whole-numberquotients. For numbers not divisible by 7, your calculator willgiveyou long decimal results that are not exact.1 Select a 2-digit number. 2 Multiply it by 7. 3 Move thedecimalpoint 1 place to the left.
  • 57. Example:1 The number chosen to multiply by 1 3/7 is 36. 2 Multiply by7:7 x 36 = 210 + 42 = 252 3 Move the decimal point 1 place totheleft: 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 by7:7 x 51 = 357 3 Move the decimal point 1 place to the left: 35.74So 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. 2Multiplyby 4: 4 x 80 + 4 x 6 = 320 + 24 = 344 3 Move the decimal pointone 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. 2Multiplyby 2: 2 x 624 = 1248 3 Multiply by 2: 2 x 1248 = 2400 + 96 =24964 Move the decimal point one place to the left: 249.6 5 So 624divided by 2 1/2 = 249.6.Multiply by 4 when this is easy; otherwise use two steps andmultiply by 2 twice.Dividing a 2-digit number by 2 1/31 Select a 2-digit number. 2 Multiply by 3. 3 Divide the resultby 7.Example:1 he 2-digit number chosen to divide by 2 1/3 is 42. 2 Multiplyby 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. 2Multiply by 3:3 x 73 = 219 3 Divide by 7: 219/7 = 31 2/7 4 So 73 divided by 21/3 = 31 2/7.If the number chosen is divisible by 7, the quotient will be awholenumber. If the number is not divisible by 7, a calculator userwillget 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. 2Multiply 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. 2Multiply by 3:3 x 61 = 183 3 Divide by 8: 183/8 = 22 7/8 4 So 61 divided by 22/3 = 22 7/8.Using this method, your answers will be exact. Those usingcalculators will only get approximations.1 Select a 2-digit number. 2 Mltmultiiply the number by 2. 3Divide theproduct by 7.Example:1 The 2-digit number chosen to divide by 3 1/2 is 42. 2Multiply by 2: 2 x42 = 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. 2Multiply by 2:
  • 59. 2 x 61 = 122 3 Divide by 7: 122/7 = 17 3/7 4 So 61 divided by 31/2 = 17 3/7.If the number chosen is divisible by 7, the answer will be awhole number.For numbers not divisible by 7, a calculator will get arepeating decimal,but your fractional answer will be exact.1 Select a 2- or 3-digit number. 2 Multiply by 3. 3 Move thedecimal pointone place to the left.Example:1 The 2-digit number chosen to divide by 3 1/3 is 72. 2Multiply by 3:72 x 3 = 216 3 Move the decimal point one place to the left:21.6 4 So72 divided by 3 1/3 = 21.6.See the pattern?1 The 2-digit number chosen to divide by 3 1/3 is 48. 2Multiply by 3:48 x 3 = 120 + 24 = 144 3 Move the decimal point one place tothe left:14.4 4 So 48 divided by 3 1/3 = 14.4.After practicing, choose larger numbers. Insist on exactanswers (youwont get an exact answer if you divide by 3.3 using acalculator).Multiply from left to right in steps and impress your friendswithyour mental powers.Dividing a 2-digit number by 3751 Select a 2-digit number. 2 Multiply by 8. 3 Divide theproduct by3 (express remainder as a fraction). 4 Move the decimal pointthree places to the left.Example:
  • 60. 1 The number chosen 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 Movethe decimal point 3 places to the left: .0853 1/3 5 So 32divided 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 Movethe decimal point 3 places to the left: .162 2/3 5 So 61 dividedby 375 = .162 2/3.72 3 Divide by 9:72/9 = 8 4 So 36divided by 4 1/2 =8.For numbers notdivisible by 9, yourcalculator will getarepeating decimal,but your fractionalanswer will beexact.Dividing a 2-digit number by6251 Select a 2-digit number. 2 Multiply by 8. 3 Divide theproductby 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 Multiplyby8: 8 x 65 = 480 + 40 = 520 3 Divide by 5: 520/5 = 104 4 Movethedecimal point 3 places to the left: .104 5 So 65 divided by 625= .104.See the pattern?
  • 61. 1 The 2-digit number chosen to divide by 625 is 32. 2 Multiplyby8: 8 x 32 = 240 + 16 = 256 3 Divide by 5: 256/5 = 51.2 4 Movethedecimal point 3 places to the left: .0512 5 So 32 divided by625 =.0512.Dividing a 2-digit number by 7 1/21 Select a 2-digit number. 2 Multiply it by 4 (or by 2 twice). 3Divideby 3. 4 Move the decimal point 1 place to the left.2M u lt i p lExample: y b 1 The 2-digit number chosen to multiply by 71/2 is 42. y4: 42 x 4 = 168 3 Divide by 3: 168/3 = 56 4 Move the decimalpoint1 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. 2Multiply byDividing a 2-digit number by 4 1/221 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 62.23Multiply by 2: 2 x 62 = 124 Divide by 9: 124/9 = 13 7/94 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 Movethe
  • 62. decimal point 1 place to the left: 12.4 5 So 93 divided by 7 1/2= 12.4.Practice and you will soon be cranking out these quotientswithspeed and accuracy.Dividing a 2- or 3-digit number by 16 2/31 Select a 2-digit number. (Choose larger numbers when youfeelsure about the method.) 2 Multiply by 6 (or by 3 and then 2). 3Move the decimal point two places to the left.Example:1 The 2-digit number chosen to divide by 16 2/3 is 72. 2Multiplyby 3: 3 x 72 = 216 3 Multiply by 2: 2 x 216 = 432 4 Move thedecimalpoint 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. 2Multiply by3: 3 x 212 = 636 3 Multiply by 2: 2 x 636 = 1200 + 72 = 1272 4Move thedecimal point 2 places to the left: 12.72 5 So 212 divided by16 2/3 = 12.72.Practice multiplying by 3, then by 2, and you will be able todo theseproblems quickly.Divisibility RulesWhy do these rules work? - Dr. RobDivisibilidad por 13 y por números primos (13,17,19...)-en español, de la lista SNARKFrom the Archives of the Math Forums Internet project AskDr. Mathourthanks to Ethan Dr. Math Magness, Steven Dr. MathSinnott,
  • 63. nd, for the explanation of why these rules work, Robert L.Ward (Dr. Rob).Dividing by 3Add up the digits: if the sum is divisible by three, then thenumber isas well. Examples: 1 111111: the digits add to 6 so the wholenumberis divisible by three. 2 87687687. The digits add up to 57, and5 plusseven 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 3As Kevin Gallagher wrote to Dr. MathOn 5/11/96 at 21:35:40 (Eastern Time),>Im looking for a SIMPLE way to explain to several verybright 2nd>graders why the divisibility by 3 rule works, i.e. add up allthe >digits;if the sum is evenly divisible by 3, then the number is as well.>Thanks!>Kevin GallagherThe only way that I can think of to explain this would be asfollows:Look at a 2 digit number: 10a+b=9a+(a+b). We know that 9a isdivisible by 3, so 10a+b will be divisible by 3 if and only if a+bis.Similarly, 100a+10b+c=99a+9b+(a+b+c), and 99a+9b isdivisibleby 3, so the total will be iff a+b+c is.This explanation also works to prove the divisibility by 9 test.Itclearly originates from modular arithmetic ideas, and Im notsure
  • 64. if its simple enough, but its the only explanation I can thinkof.Doctor Darren, The Math ForumCheck out our web site -http://forum.swarthmore.edu/dr.math/Dividing by 4ook at the last two digits. If they are divisible by 4, thenumber isas well. Examples:1 100 is divisible by 4. 2 1732782989264864826421834612 isdivisible by four also, because 12 is divisible by four.Dividing by 5If the last digit is a five or a zero, then the number is divisibleby 5.Dividing by 6Check 3 and 2. If the number is divisible by both 3 and 2, it isdivisible by 6 as well. Robert Rusher writes in:Another easy way to tell if a [multi-digit] number isdivisible by six . . .is to look at its [ones digit]: if it is even, and the sum ofthe [digits] isa multiple of 3, then the number is divisible by 6.Dividing by 7To find out if a number is divisible by seven, take the lastdigit,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, andubtract that from 20 to get 14. If you get an answer divisibleby 7(including zero), then the original number is divisible byseven.If you dont know the new numbers divisibility, you canapplythe rule again.Dividing by 8
  • 65. Check the last three digits. Since 1000 is divisible by 8, if thelast three digits of a number are divisible by 8, then so is thewhole number. Example: 33333888is divisible by 8; 33333886isnt. How can you tell whether the last three digits aredivisible by 8? Phillip McReynolds answers:If the first digit is even, the number is divisible by 8 if thelast two digits are. If the first digit is odd, subtract 4 fromthe last two digits; the number will be divisible by 8 if the resulting last two digits are. So, to continue the last example,33333888 is divisible by 8 because the digit in the hundredsplace is an even number, and the last two digits are 88, whichis divisible by 8. 33333886 is not divisible by 8 because thedigit in the hundreds place is an even number, but the lasttwo digits are 86, which is not divisible by 8.Dividing by 9Add the digits. If they are divisible by nine, then the numberis as well. This holds for any power of three.Dividing by 10If the number ends in 0, it is divisible by 10.Dividing by 11Lets look at 352, which is divisible by 11;the answer is 32. 3+2 is 5; another way to say thisis that 35 -2 is 33. Now look at 3531, which is alsodivisible by 11. It is not a coincidence that 353-1 is352 and 11 x 321 is 3531. Here is a generalizationof this system. Lets look at the number 94186565.First we want to find whether it is divisible by 11, but on theway we are going to save the numbers that we use: in everystep we will subtract the last digit from the other digits, thensave the subtracted amount in order. Start with9418656 - 5 = 9418651 SAVE 5 Then 941865 - 1 = 941864SAVE 1Then94186 - 4 = 94182 SAVE 4 Then 9418 - 2 = 9416 SAVE 2 Then941- 6 = 935
  • 66. SAVE 6 Then 93 - 5 = 88 SAVE 5 Then 8 - 8 = 0 SAVE 8Now write the numbers we saved in reverse order, and wehave8562415, which multiplied by 11 is 94186565.Heres an even easier method, contributed by ChisForen:Takeany 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 isthenumber. 22 - 22 = 0 so 365167484 is evenly divisible by11.See alsoDivisibility by 11 in the Dr. Math archives.Dividing by 12Check for divisibility by 3 and 4.Dividing by 13Heres a straightforward method supplied by Scott Fellows:D e lete the last digit from the given number. Then subtractninetimes the deleted digit from the remaining number. If what isleftis divisible by 13, then so is the original number.And heres a more complex method that can be extended tootherformulas:1 = 1 (mod 13)10 = -3 (mod 13) (i.e., 10 - -3 is divisible by13)100= -4 (mod 13) (i.e., 100 - -4 is divisible by 13)1000 = -1 (mod13)(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 number is divisible by 13, then so is the originalnumber.You can keep using this technique to get other formulas fordivisibility for prime numbers. For composite numbers justcheck for divisibility by divisors.Finding 2 1/2 percent of a number1 Choose a number (start with 2 digits and advance to 3 withpractice).2 Divide by 4 (or divide twice by 2). 3 Move the decimal pointone placeto the left.Example:1 If the number selected is 86: 2 Divide 86 by 4: 86/4 = 21.5 3Move thedecimal 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 So2 1/2% of 648 = 16.2.Practice dividing by 4, or by 2 twice, and you will be able tofind theseanswers faster than with a calculator.Finding 5 percent of a number1 Choose a large number (or sum of money). 2 Move thedecimalpoint 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 thedecimal pointone 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 the amount of money selected is $4500: 2 Move thedecimal pointone place to the left.: 450 3 Divide by 2: 450/2 = 225 4 So 5%of $4500 =$225.Finding 15 percent of a number1 Choose a 2-digit number. 2 Multiply the number by 3. 3Divide 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 3Divideby 2: 129/2 = 64.5 4 Move the decimal point one place to theleft: 6.455 So 15% of 43 = 6.45.See the pattern?1 If the number selected is 72: 2 Multiply by 3: 3 x 72 = 216 3Divide by2: 216/2 = 108 4 Move the decimal point one place to the left:10.85S o15%of72=10.8.Finding 45 percent of a number1 Choose a 2-digit number. 2 Multiply the number by 9. 3Divide 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 3Divide by 2: 324/2 = 163 4 Move the decimal point one placeto the left: 16.2 5 So 45% of 36 = 16.2.See the pattern?
  • 69. If the number selected is 52: Multiply by 9: 9 x 52 = 450 + 18 =468 Divideby 2: 468/2 = 234 Move the decimal point one place to theleft: 23.4 So45% of 52 = 23.4.Finding 55 percent of a numbern 1 Choose a 2-digit number. e 2 Multiply the number by 11.(Add digitsfrom right to left - see examples). x 3 Divide by 2. t 4 Move thedecimalpoint one place to the left. di Example: git 1 If the number selected is 81: t 2 Multiply by 11: 11 x 81 =891 o rightdigit is 1 l eft is 1 + 8 = 9 last digit to left is 83 Divide by 2: 891/2 = 445.5 4 Move the decimal point oneplace to theleft: 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 leftis 9 + 5= 14 (use the 4 and carry 1)last digit to left is 5 + 1 = 62 Divide by 2: 649/2 = 324.53 Move the decimal point one place to the left: 32.454 So 55% of 59 = 32.45.Why do the divisibility "rules" work?The Power of Modulo ArithmeticModulo arithmetic is a powerful tool that can be used to testfordivisibility by any number. The great disadvantage of usingmodulo arithmetic to test for divisibility is the fact that it isusuallya slow method. Under some circumstances, however, the
  • 70. application of modulo arithmetic leads to divisibility rulesthat canbe used. These rules are important to mathematics becausetheysave 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. Thispage will show you how some of these common divisibilityrules are connected to modulo arithmetic.Applying the Rules of Modulo ArithmeticThe rules of modulo arithmetic state that the number N isdivisibleby some number (P) if the above expression is also divisibleby Pafter the base (10) is replaced by the remainder of 10 dividedby P.In compact notation, this remainder is denoted by (10 modP). Thus, N is divisible by P if the following expression is alsodivisible by P:[Dn * (10 mod P)^(n-1)] + ... + [D2 * (10 mod P)^1] + [D1 * (10mod P)^0].Origin of Divisibility Rules for 3, 9, and 11D We can now see how the divisibility rules for P = 3, 9, and11 arerooted in modulo arithmetic. nConsider the case where P = 3. .Because 10 mod 3 is equal to 1, any number is divisible by 3if thefollowing expression is. also divisible by 3: .-[Dn * (1)^n-1] + ... + [D2 * (1)^1] + [D1* D2(1)^0]. Because 1 raised to any power is equal to 1, the above
  • 71. +expression can be simplified as: Dn + ... + D2 + D1 + D0,Dwhich is equal to the sum of digits in the number. Thusmoduloarithmetic allows us to 1state that any number is divisible by 3 if the sum of its digits isBecause 10 mod 9 is also 1, any number is divisible by 9 isthesum of its digits are alsodivisible by 9. Thus, the divisibility rules for 3 and 9 comedirectlyfrom modulo narithmetic.n)Finally, consider the case where P = 11. aBecause 10 mod 11 is equal to -1, any number is divisible by11if the n following expression is also divisible by 11: dD[Dn * (-1)^n-1] + ... + [D2 * (-1)^1] + [D1* (-1)^0].nBecause -1 raised to any even power is equal to 1 and -1raised toany odd +power is equal to -1, the above expression becomesa summation of the digits involving alternated signs: .. . - D2 + D1 (for odd n).This "alternating" summation rule for P = 11 is well knownand has two versions. The first version states a number isdivisible by 11 if the alternating sum of the digits is alsodivisilbeby 11 (i.e., the alternating sum is a muliple of 11). The secondversion states that a number is divisible by 11 if the sum ofeveryother digit starting with the rightmost digit is equal to thesum of
  • 72. every other digit starting with the second digit from the right(i.e.,the alternating sum is 0). Both of these versions comedirectlyfrom the rules of modulo arithmetic.Bases Other Than 10v Up to this point, we have only considered numbers writteninbase 10. A number can, however, be i written in any base (B).Such a number can be expressed by the followingsummation: git 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 isused. These rules tell us that N is divisible by P if the followingexpression is also divisible by P: [Dn * (B mod P)^n-1] + ...+ [D2 * (B mod P)^1] + [D1* (B mod P)^0]. Most bases otherthan 10 are difficult to use, but basesthat are powers of 10 can easily beused 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 andgrouping the digits in pairs of two. bEach grouping oftwo digits is considered a single "digit" when the numberis ywritten in base-100. Written in base-100, the number is1 23 34 57 where 57, 34, 123, and 1 are considered single"digits." In a similar fashion, this number written 1inbase-1000 is 1 233 457 where 457, 233, and 1 are consideredsingle "digits." .Once the base-10 digits are grouped to formthe "digits," the above expression Bcan be used to test fordivisibility. Consider P = 7.Use the base, B = 1000. Because 1000 mod 7 = -1, the
  • 73. alternating summation rule a (used for P = 11 in base-10) canbe applied. Before applying this rule, the base-10 pdigitsmustbe properly grouped to form base-1000 "digits." pe Stryker explains it this way: When you are multiplying by 9,onyour fingers (starting with your thumb) count the number youaremultiplying by and hold down that finger. The number offingersbefore the finger held down is thefirst digit of the answer and the number of finger after thefingerheld down is the second digit of the answer.Example: 2 x 9. your index finder is held down, your thumb isbefore, representing 1, and there are eight fingers after yourindexfinger, representing 18.● Polly Norris suggests: When you multiply a number times9, countback one from that number to get the beginning of yourproduct. (5 x 9: one less than 5 is 4). To get the rest of your answer,just thinkof the add fact families for 9:1+8=92+7=93+6=94+5=98+1=97+2=96+3=95+4=95 x 9 = 4_. Just think to yourself: 4 + _ = 9 because the digitsin yourproduct 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 teachthe "ninesin multiplication to my third graders and they learn them inone lesson!
  • 74. Tamzo explains this a little differently:1 Take the number you are multiplying 9 by and subtract one.Thatnumber is the first number in the solution. 2 Then subtractthatnumber from nine. That number is the second number of thesolution.Examples:4 * 9 = 361 4-1=32 9-3=63 solution = 368 * 9 = 721 8-1=7 2 9-7=2 3 solution = 725 * 9 = 451 5-1=4 2 9-4=5 3 solution = 45● Sergey writes in: Take the one-digit number you aremultipling bynine, and insert a zero to its right. Then subtract the originalnumberfrom it.For example: if the problem is 9 * 6, insert a zero to the rightof the six,then subtract six: 9 * 6 = 60 - 6 = 54Multiplying a 2-digit number by 11● A tip sent in by Bill Eldridge: Simply add the first andseconddigits and place the result between them.Heres an example using 24 as the 2-digit number to bemultipliedby 11: 2 + 4 = 6 so 24 x 11 = 264.This can be done using any 2-digit number. (If the sum is 10or greater, dont forget to carry the one.)Multiplying any number by 11
  • 75. ● Lonnie Dennis II writes in:Lets say, for example, you wanted to multiply 54321 by 11.First, letslook at the problem the long way...54321x 1154321 + 543210=597531Now lets look at the easy way...11 x 543214432=51+5 +3 +2 +1531Do you see the pattern? In a way, youre simply adding thedigit towhatever comes before it. But you must work from right toleft.The reason I work from right to left is that if the numbers,whenadded together, sum to more than 9, then you havesomethingto carry over. Lets look at another example... 11 x 9527136Well,we know that 6 will be the last number in the answer. So theanswernow is???????6.Calculate the tensplace: 6+3=9, so nowwe know that theproduct has the form??????96. 3+1=4, sonow we know that the
  • 76. product has the form?????496. 1+7=8, so????8496. 7+2=9, so???98496. 2+5=7, so??798496. 5+9=14. Heres where carrying digits comes in: wefill in the hundred thousands place with the ones digit of thesum5+9, and our product has the form ?4798496. We will carrythe extra10 over to the next (and final) place. 9+0=9, but we need toadd theone carried from the previous sum: 9+0+1=10. So the productis104798496. Calculation Tips & TricksMultiplication TipsMultiplying by five● Jenny Logwood writes: Here is an easy way to find ananswer toa 5 times question. If you are multiplying 5 times an evennumber:halve the number you are multiplying by and place a zeroafter thenumber. Example: 5 x 6, half of 6 is 3, add a zero for ananswer of30. Another example: 5 x 8, half of 8 is 4, add a zero for ananswerof 40.If you are multiplying 5 times an odd number: subtract onefromthe number you are multiplying, then halve that number andplacea 5 after the resulting number. Example: 5 x 7: -1 from 7 is 6,half of6 is 3, place a 5 at the end of the resulting number to producethe
  • 77. number 35. Another example: 5 x 3: -1 from 3 is 2, half of 2 is1,place a 5 at the end of this number to produce 15.● Doug Elliott adds: To square a number that ends in 5,multiplythe tens digit by (itself+1), then append 25. For example: tocalculate 25 x 25, first do 2 x 3 = 6, then append 25 to thisresult;the answer is 625. Other examples: 55 x 55; 5 x 6 = 30,answeris 3025. You can also square three digit numbers this way, bystarting 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,holdup ten fingers - if the problem is 9 x 8 you just put down your8 fingerand theres your answer: 72. (If the problem is 9 x 7 just putdownyour 7 finger: 63.)� Laurie Stryker explains it this way: When you aremultiplying by9, on your fingers (starting with your thumb) count thenumber youare multiplying by and hold down that finger. The number offingersbefore the finger held down is thefirst digit of the answer and the number of finger after thefingerheld 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 yourindexfinger, representing 18.Norris suggests: When you multiply a number times 9, countbackone 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, justthinkof 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 =95+4=95 x 9 = 4_. Just think to yourself: 4 + _ = 9 because the digitsinyour product n always add up to 9 when one of the factors is9.Therefore, 4 + 5 = 9 and your answer is 45! I use this methodtoeach the "nines" in multiplication to my third e graders andtheylearn them in one lesson!Tamzo explains this a little differently:1. Take the number you are multiplying 9 by and subtractone.That number is the first e number in the solution. 2 Thensubtractt that number from nine. That number is the second numberof thesolution.Examples:n 4 * 9 = 36 g1 4-1=3 2 9-3=6 2 3 solution = 36 48 * 9 = 72as 1 8-1=7 2 9-7=2 t 3 solution = 72 h
  • 79. e5 * 9 = 452 -d 1 5-1=4 i 2 9-4=5 g3 solution = 45 itMultiplying a 2-digit number by 11n● A tip sent in by Bill Eldridge: Simply add the first andseconddigits and place the result ubetween them. m b er to be multiplied by 11: 2 + 4 = 6 so 24 x11 = 264.This can be done using any 2-digit number. (If the sum is 10or greater, dont forget to carry the one.)Multiplying any number by 11● Lonnie Dennis II writes in:Lets say, for example, you wanted to multiply 54321 by 11.First,lets look at the problem the long way...54321x 1154321 + 543210= 597531 Now lets look at the easy way... 11 x 54321=51+5 +3 +2 +1Do you see the pattern? In a way, youre simply adding thedigitto whatever comes before it. But you must work from right toleft.The reason I work from right to left is that if the numbers,whenadded together, sum to more than 9, then you havesomethingto carry over.
  • 80. Lets look at another example... 11 x 9527136 Well, we knowthat6 will be the last number in the answer. So the answer now is6. Calculate the tens place: 6+3=9, so now we know that theproduct has the form ??????96. 3+1=4, so now we know thatthe product has the form?????496. 1+7=8, so ????8496.7+2=9, so ???98496. 2+5=7, so ??798496. 5+9=14. Hereswhere carrying digits comes in: we fill in the hundred thousandsplacewith the ones digit of the sum 5+9, and our product has theform ?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 previoussum:9+0+1=10.So the product is 104798496.Multiplying a 3-digit number by 991 Select a 3-digit number. 2 Subtract the 1st digit plus 1 fromthe number. X X X _ _3 Subtract the last two digits of the number from 100. _ _ _ XXExample:1 The 3-digit number chosen to multiply by 99 is 274. 2Subtractthe 1st digit + 1 from the number: 274 - 3 = 271 : 2 7 1 _ _3 Subtract the last two digits from 100: 100 - 74 = 26: _ _ _ 26.4 So 274 x 99 = 27126.See the pattern?1 The 3-digit number chosen to multiply by 99 is 924. 2Subtract 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 1431 Select a 3-digit number. 2 Repeat the 3 digits to produce a6-digitnumber. 3 Divide by 7.Example:1 The 3-digit number chosen to multiply by 143 is 123. 2Repeat 3digits: 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 So143 x 765 = 109,395.Multiplying a 4-digit number by 991 Select a 4-digit number. 2 Subtract the first two digits plus1 from thenumber. X X X X _ _3 Subtract the last two digits of the number from 100.____XXEasier example:1 Choose a 4-digit with digits from smaller to larger: 2368. 2The firsttwo digits will be the same: 2 3 _ _ _ _. 3 Subtract the firsttwo digits+ 1 from the last two digits: 23 + 1 = 24, 68 - 24 = 44: _ _ 4 4 4Subtractthe last two digits from 100: 100 - 68 = 32: _ _ _ _ 3 25 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: _ _ _ _ 88 3 So3512 x 99 = 347688.Start out with easier examples and graduate to more difficultones.Remember to subtract from left to right in increments.Multiplying a repeating 9-digit number by 3, then dividing by123456791 Choose a repeating 9-digit number (one whose numbersare 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: 3x 9 = 27. 3Multiply 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: 9x 6 = 54. 3Multiply 3: 54 x 3 = 162. 4 So 666666666 x 3 / 12345679 = 162.Change the second multiplier to other numbers and createmany variationsof this exercise.Multiplying two mixed numbers1 Select a mixed number. (1 2/3, 5 1/8, etc.) 2 Select amultiplier using thesame whole number and a fraction that sums to 1 with thefirst numbersfractional part. 3 The whole number in the answer will be thewholenumber times the next number. 4 To find the fraction,multiply the twofractional parts.Example:
  • 83. 1 The first mixed number chosen is 7 1/4. 2 Select 7 3/4 asthe multiplier(same whole number, fraction that sums to 1 with the firstfraction.3 The products whole number will be 7 x 8 (next number) =56. 4 Theproducts 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 asthemultiplier (same whole number, raction that sums to 1 with the first fraction. 53 The products whole number will be 8 x 9 (next number) =72. 4The products 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 asthemultiplier (same whole number, fraction that sums to 1 withthefirst fraction.3 The products whole number will be 20 x 21 (next number) =420.4 The products fraction will be 1/8 x 7/8 = 7/64. 5 So 20 1/8 x207/8 = 420 7/64.Multiplying a 4-digit number by 137, then by 73; subtract12001 Choose a 4-digit number (one whose numbers are all thesame).2 Multiply by 137 and multiply the product by 73. 3 Thesecond
  • 84. product is an 8-digit number: the original number and arepeat ofthe digits. 4 Subtract 1200.Example:1 If the first number is 6241: 2 The answer to the twomultiplicationsis 62416241 3 Subtract 1200: 62416241 - 1200 = 62415041 4So6241 x 137 x 73 = 62,415,041.See the pattern?1 If the first number is 7834: 2 The answer to the twomultiplicationsis 78347834 3 Subtract 1200: 78347834 - 1200 = 78346634 4So 7834x 137 x 73 = 78,346,634.You can expand this exercise by changing the final step toanothercalculation: add or subtract a number that is easy to handle.In thisway you can create many newMultiplying a 5-digit number by 9091, then by 11; subtract1200act1200.Example:1 If the first number is 34986: 2 The answer to the twomultiplications is3498634986 3 Subtract 1200: 3498634986 - 1200 = 34986337864 So 34986x 9091 x 11 = 3,498,633,786.See the pattern?1 If the first number is 22854: 2 The answer to the twomultiplications i
  • 85. 2285422854 3 Subtract 1200: 2285422854 - 1200 = 22854216544 So22854 x 9091 x 11 = 2,285,421,654.You can expand this exercise by changing the final step toanothercalculation: add or subtract a number that is easy to handle.In thisway you can create many newMultiplying a multi-digit sequence of numbersby 8 and adding the last digit1 Choose a number made up of consecutive digits beginningwith 1(123, 1234 etc.). 2 Multiply it by 8 and add the last digit of theoriginal number. 3 The answer will have the same number ofdigits as the first number selected; the digits will bedecreasing digits starting with 9.Example:1 If the number selected is 12345: 2 Multiply by 8, add 5: 3Theanswer is 98765 (5 digits). 4 So 12345 multiplied by 8 plus 5is 98765.See the pattern?1 If the number selected is 12345678: 2 Multiply by 8, add 8: 3The answeris 98765432 (8 digits). 4 So 12345678 multiplied by 8 plus 8 is98765432.More than one example will probably give away the secret,but thisought to be good for one-shot tries. Multiply a multi-digitsequenceof numbers1 Choose a 5-digit number.2 Multiply by 9091 and multiply the product by 11.
  • 86. 3 The second product is a 10-digit number: the originalnumber anda repeat of the digits.by 8, add a number, subtract 201 Choose a number made up of consecutive digits beginningwith 1(123, 1234 etc. - up to 9 digits). 2 Multiply it by 8. 3 Add thelast digitof the original sequence. The answer will be a decliningsequencebeginning with 9 and the same length as the originalsequence.4 Subtract 20.Example:1 If the number selected is 123: 2 Multiply by 8, add 3(descendingsequence beginning with 9 and 3 digits long): 987 3 Subtract20:987 - 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 thelastsubtraction is 987654 (descending sequence from 9 with 6digits)3 Subtract 20: 987654 - 20 = 987634 4 So 123456 multiplied by8 + 6 - 20 is 987634.Extend the number of examples by changing the last step –orsubtract or add different numbers.Multiply the sum and difference of a 2-and a 1-digit numberrence.Exa
  • 87. mple:1 If the numbers 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).d2 Add the two numbers if3 Find their difference (subtract the smaller from the larger). f4 Multiply the sum and the difference by squaring the twonumbers and finding the eMultiplying a 2-digit number by 1 1/61 Select a 2-digit number. 2 Multiply by 7. 3 Divide the resultby 6.Example:1 The 2-digit number chosen to multiply by 1 1/6 is 34. 2Multiply by7: 7 x 34 = 210 + 28 = 238 3 Divide by 6: 238/6 = 39 4/6 4 So 34x11/6 = 39 2/3.See the pattern?1 If the 2-digit number chosen to multiply by 1 1/6 is 57: 2Multiplyby 7: 7 x 57 = 350 + 49 = 399 3 Divide by 6: 399/6 = 66 3/6 4 So57 x
  • 88. 1 1/6 = 66 1/2.Multiplying a 2-digit number by 1 1/82D 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: 2Multiplyby 9: 9 x 32 = 270 + 18 = 288 3 Divide by 8: 288/8 = 36 4 So 32x 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 = 797/8.Multiplying a 2-digit number by 1 1/91 Select a 2-digit number. 2 Add a zero to the number. 3Dividethe result by 9.Example:1 The 2-digit number chosen to multiply by 1 1/9 is 32. 2 Addzero: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: 2Add 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 theiranswers will be repeating decimals. Your answer will beexact.Multiplying a 2-digit number by 1 1/41 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: 2Add azero: 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 the number to be multiplied by 1 1/4 is 72: 2 Add a zero:720 3Divide by 8: 720/8 = 90 4 So 72 x 1 1/4 = 90.Multiplying a 2-digit number by 1 5/61 Select a 2-digit number. 2 Multiply by 11. 3 Divide the resultby 6.Example:1 The 2-digit number chosen to multiply by 1 5/6 is 34. 2Multiply by 11: 11 x 34 = 374 (Add the digits left to right. The last digit is 4.The nextdigit 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: 2Multiply by11: 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 by11.Multiplying a 2-digit number by 1 3/41 Multiply the number by 7. 2 Divide by 4.Example:1 If the number chosen to multiply by 1 3/4 is 42: 2 Multiplyby 7:7 x 42 = 280 + 14 = 294 3 Divide by 4: 294/4 = 73 2/4 4 So 42 x1 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 x1 3/4 = 50 3/4.Practice, and these products will be fast and accurate.Multiplying a repeating 3-digit number by 2, then dividing by37
  • 90. 1 Choose a repeating 3-digit number (one whose numbersareall 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 3/41Multiply 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 Multiplyby 11.Add digits from right to left: _ _ 8 8 + 7 = 15 7 + 1 (carry) = 878 * 11 = 8583 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 by11:11 x 27 = 297 3 Divide by 4: 297/4 = 74 1/4 4 So 27 x 2 3/4 = 741/4.Multiplying a repeating 3-digit number by 5, then dividing by371 Choose a repeating 3-digit number (one whose numbersare allthe same). 2 Add its digits (or multiply one of them by 3). 3Multiply by 5.Example:
  • 91. 1 If the first 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 bymultiplying byother numbers. You can produce many new exercises in thismanner.Multiplying a 2-digit number by 7 1/21 Multiply by 3. 2 Divide by 4. 3 Add a zero or move thedecimal pointone place to the right.Example:1 If the 2-digit number chosen to multiply by 7 1/2 is 64: 2Multiply by3: 3 x 64 = 180 + 12 = 192 3 Divide by 4: 192/4 = 48 4 Addzero: 480 5So 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 decimalpoint one place to the right: 697.5 5 So 93 x 7 1/2 = 697.5Multiplying a 2-digit number by 12 1/21 Select a 2-digit number . 2 Divide it by 8 (or by 2 threetimes). 3 Movethe decimal point 2 places to the right.Example:1 The 2-digit number chosen to multiply by 12 1/2 is 28. 2Divide by 8: 28/8
  • 92. = 3.5 3 Move the decimal point 2 places to the right: 350 4 So28 x 12 1/2 =350.See the pattern?1 If the 2-digit number chosen to multiply by 12 1/2 is 58: 2Divide by 8by 2 three times): 58/2 = 29 29/2 = 14.5 14.5/2 = 7.253M o v e t h e d e c i m a l p o i n t 2 places to the right: 725 4So 58 x12 1/2 = 725.Multiplying a 2-digit number by 151 Select a 2-digit number. 2 Add a zero to it. 3 Divide it by 2. 4Add thenumber obtained by dividing by 2 to the last number.Example:1 The 2-digit number chosen to multiply by 15 is 62. 2 Add azero: 620.3 Find half (divide by 2): 620/2 = 310. 4 add: 620 + 310 = 930. 5So 62 x 15 =930.See the pattern?1 If the number to be multiplied by 15 is 36: 2 Add a zero:360. 3 Findhalf (divide by 2): 360/2 = 180. 4 add: 360 + 180 = 540. 5 So 36x 15 = 540.Multiplying a 2-digit number by 181 Select a 2-digit number. S 2 Multiply it by 2. e 3 Add onezero. e 4Subtract the number obtained by multiplying by 2 from thelast number. th Example: ep 1 The 2-digit number chosen to multiply by 18 is 28. a 2 28x 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. rn?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 ineasy increments:920 - 100 = 820, 820 + 8 = 828)5 So 46 x 18 = 828. Practice this procedure, especiallyselecting the easiestways to subtract.Multiplying a 2-digit number by 221 Select a 2-digit number. 2 Multiply it by 2. 3 The last digitwill be thesame. _ _ _ 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 toleft.6 + 5 = 11 _ _ 1 _ 5 + 1 + 1 (carry) = 7 _ 7 _ _ 5 The first digit isthesame. 1 _ _ _ 6 So 22 x 78 = 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. Thenyou will be able to multiply by 22 in your head with ease.Multiplying a 2- or 3-digit number by 251 Select a 2- or 3-digit number. (Choose larger numbers whenyou feel sure about the method.) 2 Divide by 4 (or by 2 twice).
  • 94. 3 Add 2 zeros (or move the decimal point 2 places to theright).Example:1 The 2-digit number chosen to multiply by 25 is 78. 2 Divideby 2 twice: 78/2 = 39, 39/2 = 19.5 3 Move the decimal point 2places 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 Divideby 2 twice: 258/2 = 129, 129/2 = 64.5 3 Move the decimal point2 places to the right: 6450 4 So 258 x 25 = 6450Multiplying a 2-digit number by 271 1 Select a 2-digit number . 8 2 Multiply it by 3. 9 3 Add onezero.0 4 Subtract the number obtained by multiplying by 3 from thelast number. -1 Example: 8 9 1 The 2-digit number chosento multiply by 27 is 42. = 2 3 x 42 = 120 + 6 = 126 (multiplyby 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 01 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 – 189 = 1701 0 (Subtract left to right in steps: -9 = 1790 - 80 - 9 =1710 - 9 = 1701or 1890 - 190 + 1 = 1700 + 1 = 1701.5. So 63 x 27 = 1701.Multiplying a 2-digit number by 331 Select a 2-digit number. 2 Multiply it by 3. 3 Add thedigits from right to left (see examples, below).Example:1 The 2-digit number chosen to multiply by 33 is 38. 2Multiply by 3: 3 x 38 = 90 + 24 = 114 3 The last digit is4: _ _ _ 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 the number to be multiplied by 33 is 82: 2 Multiply by3: 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 afriend and BEAT THE CALCULATOR!le:1 The 2-digitnumber chosen tomultiply by 33 1/3is 48.2 Add two zeros: 4800 3 Divide by 3: 4800/3 = 1600 4 So48 x 33 1/3 = 1600.See the pattern?1 The 2-digit number chosen to multiply by 33 1/3 is 92. 2Add two zeros: 9200 3 Divide by 3: 9200/3 = 3066 2/3. 4So 92 x 33 1/3 = 3066 2/3.Multiplying a 2-digit number by 35Select a 2-digit number.1 Multiply it by 7. 2 Divide by 2. 3 Add a zero.Multiplying a 2-digit number by 33 1/312Select 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 x28 = 140 + 56 = 196. 3 Divide by 2: 196/2 = 98. 4 Add onezero: 980. 5 So 35 x 28 = 980.See the pattern?1 The 2-digit number chosen to multiply by 35 is 68. 2 7 x68 = 420 + 56 = 476. 3 Divide by 2: 476/2 = 238. 4 Add onezero: 2380. 5 So 35 x 68 = 2380.Practice and you will be able to give these products easilyand accurately.
  • 96. Multiplying a 2-digit number by 361S1. Select a 2-digit number. 2 Multiply it by 4. 3 Add one zero. 4 Subtract the number obtained by multiplying by 4 from thelast 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 toproduce these products of 36 and 2-digit numbers quicklyand accurately.Multiplying a 2-digit number by 37 1/21T h e 21 Select a 2-digit number. -2 Multiply it by 3. d 3 Divide by 8. i4 Move the decimal point two places to the right. git Example: number chosen to multiply by 37 1/2 is 68. 2 Multiply by 3. 3 x68= 180 + 24 = 204 3 Divide by 8. 204/8 = 25.5 4 Move thedecimalpoint two places to the right: 2550 5 So 37 1/2 x 68 = 2550.See the pattern?
  • 97. 1 The 2-digit number chosen to multiply by 37 1/2 is 28. 2Multiplyby 3. 3 x 28 = 60 + 24 = 84 3 Divide by 8. 84/8 = 10.5 4 Movethedecimal 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 441 Select a 2-digit number. 2 Multiply it by 4. 3 Add the digitsfromright 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 144right toleft: 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 276righto 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 451 Select a 2-digit number. 2 Multiply it by 5. 3 Add a zero. 4Find thedifference between these two numbersExample:1 The 2-digit number chosen to multiply by 45 is 54. 2 5 x 54= 270.
  • 98. 3 Add one zero: 2700. 4 Subtract: 2700 - 270 = 2430 (subtractleft toright: 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 = 135Think 5 x27 = (5 x 20) + (5 x 7) = 100 + 35 = 135.3 Add one zero: 1350. 4 Subtract: 1350 - 135 = 1215 (subtractleft toright: 1350 - 100 - 30 - 5 =1250 - 30 - 5 = 1220 - 5 = 1215)5. So 27 x 45 = 1215.Practice - especially the subtraction - and you will becomeadept at finding 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 easyway to dothis is to take half of it (divide by 2), then add two 0 zeros.) 0 3Add 1/10(one-tenth) of the product. (Take off the last zero of theproduct andadd this + number to the product.) 1 4 Example: 0 = 1 The 2-digit numberchosen to multiply by 55 is 28. 1 2 50 x 28 = 1400 (half of 28 is14; add2 zeros). 5git number chosen to multiply by 63 is 28. 2 7 x 28 = 196 3Add 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 4Subtract: 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 661 Select a 2-digit number. 2 Multiply it by 6. 3 Add the digitsfromright to left (see examples, below).Example:1 The 2-digit number chosen to multiply by 66 is 54. 2Multiply by6: 6 x 54 = 300 + 24 = 324 3 The last digit is 4: _ _ _ 4 4 Addright to 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: 6x 92= 540 + 12 = 552 3 The last digit is 2: _ _ _ 24. 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.Challengea friend and BEAT THE CALCULATOR!Multiplying a 2-digit number by 72m 1 Select a 2-digit number. u 2 Multiply it by 8. lt 3 Add onezero. i 4Subtract the first number from the second. p li Example: c a 1The 2-digit number chosen to multiply by 72 is 54. ti 2 Multiplyby 8: 54 x 8 = 432 o (Think: (8 x 50) + (8 x 4) = 400 + 32 = 432) n b3 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)25 So 54 x 72 = 3888.See the pattern?1 The 2-digit number chosen to multiply by 72 is 21. 2Multiply 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 andyou will become perfect in yourMultiplying a 2- or 3-digit number by 751 Select a 2- or 3-digit number. 2 Multiply it by 3. Th 3 Addtwozeros. e 4 Divide by 4. 3dig Example: itnu The 2-digit number chosen to multiply by 75 is 62. mbMultiply by 3: 3 x 62 = 186 er Add two zeros: 18600 ch Divideby 4 (or by 2 twice): 18600/2 = 9300, 9300/2 = 4650 os So 75 x62 = 4650. ento See the pattern? muThe 3-digits number ltiply by 75 is 136. 1 Multiply by 3: 3 x136 =390 + 18 = 408 2 Add two zeros: 40800 3 Divide by 4 (or by 2twice): 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 divideby 2 twice if division by 4 is not easyMultiplying a 2-digit number by 771 Select a 2-digit number. 2 Multiply it by 7. 3 Add the digitsfromright to left (see examples, below).
  • 101. Example:1 The 2-digit number chosen to multiply by 77 is 42. 2Multiplyby 7: 7 x 42 = 280 + 14 = 294 3 The last digit is the same: _ _ _44 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: 7x 84= 560 + 28 = 588 3 The last digit is the same: _ _ _ 8 4 Addright toleft. 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 811 Select a 2-digit number. 2 Multiply it by 9. 3 Add one zero. 4Subtract the first number from the second.Example:1 The 2-digit number chosen to multiply by 81 is 31. 2Multiply by9: 9 x 31 = 279 3 Add one zero: 2790 4 Subtract: 2790 - 279 =2511 (Think: 2790 - 200 - 70 - 9 = 2590 - 70 - 9 = 2520 - 9 = 2511) 5So31 x 81 = 2511.See the pattern?1 The 2-digit number chosen to multiply by 81 is 68. 2Multiply by 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 toright. With that skill you will be able to multiply by 81 quickly.Multiplying a 2-digit number by 881 1 Select a 2-digit number. 2 Multiply it by 8. T 3 Add the digitfrom right to left. h e Example: 2 -1 the 2 digit number chosen to multiply by 88 is 43. 2 Multiplyby8: 8 x 43 = 320 + 24 = 344.3 The last digit is the same: _ _ _ 4 4 Add the digits in 344right toleft: 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. 42Multiplyby 8: 8 x 82 = 640 + 16 = 656.3 The last digit is the same: _ _ _ 6 5 Add the digits in 656righto 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 991 Select a 2-digit number. 2 Add two zeros to the number. 3Subtract the original number from the second number.Example:1 The 2-digit number chosen to multiply by 99 is 79. 2 Addtwozeros: 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?hosen to multiply by 99 is 42. 2 Add two zeros: 4200. 3Subtracthe 2-digit number: 4200 - 42 = 4200 - 40 - 2 = 4160 - 2 = 4158.4 So 42 x 99 = 4158.Multiplying a 2-digit number by 1011 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 1011 Select a 3-digit number . 2 The sum of the first and thirddigitswill be the middle digit: _ _ X _ _. 3 The first two digits plusthecarry will be the first digits: X X _ _ _. 4 The last two digits oftheExample:1 The number chosen is 318. 2 3 + 8 = 11 (sum of first andthirddigits): _ _ 1 _ _ (keep carry, 1)3 31 + 1 = 32 (first two digits plus carry): 3 2 _ _ _. 4 The lasttwodigits 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 andthirddigits): _ _ 5 _ _ (keep carry, 1) 3 72 + 1 = 73 (first two digitsplus 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 1/21 Select a 2-digit number. 2 Multiply by 3. 3 Divide by 2.Example:
  • 104. 1 The 2-digit number chosen to multiply by 1 1/2 is 63. 2Multiplyby 3: 3 x 63 = 180 + 9 = 189 3 Divide by 2: 189/2 = 94 1/2 4 So63 x1 1/2 = 94 1/2.See the pattern?1 If the 2-digit number chosen to multiply by 1 1/2 is 97: 2Multiplyby 3: 3 x 97 = 270 + 21 = 291 3 Divide by 2: 291/2 = 145 1/2 4So 97 x 1 1/2 = 145 1/2.Remember, if you use a calculator you will get a decimalapproximation, whereas the fractional form will be exact.1Multiplying a 2-digit number by 1 1/3Select 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 by4: 4 x 83 = 320 + 12 = 332 Divide by 3: 332/3 = 110 2/3 So 83 x1 1/3 = 110 2/3.Remember, if you use a calculator you will get a decimalapproximation, whereas the fractional form will be exact.Multiplying a 2-digit number by 1 1/5
  • 105. 5 Select a 2-digit number. 6 Multiply the number by 6. 7Divide the product by 5.Example:8 The 2-digit number chosen to multiply by 1 1/5 is 34. 9Multiply by6: 6 x 34 = 180 + 24 = 204 10 Divide the product by 5: 204/5 =40.8 11 So 34 x 1 1/5 = 40.8.See the pattern?12 The 2-digit number chosen to multiply by 1 1/5 is 61. 13Multiply by 6: 6 x 61 = 366 14 Divide the product by 5: 366/5 = 73.2 15So 34 x16 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 azero.Example:19 The 2-digit number chosen to multiply by 1 2/3 is 78. 20Divide by 2:78/2 = 39 21 Divide by 3: 39/3 = 13. (39 is divisible by 3because thesum of its digits is divisible by 3.) 22 Add a zero: 130. 23 So78 x 1 2/3 = 130.See the pattern?24 If the 3-digit number chosen to multiply by 1 2/3 is 315: 25Divide by3: 315 / 3 = 105. (315 is divisible by 3 because the sum of itsdigits isdivisible by 3.) 26 Divide by 2: 105/2 = 52.5. 27 Move thedecimal pointone 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 thedigits isdivisible by
  • 106. Multiplying a 2-digit number by 12529 Select a 2-digit number. 30 Divide the number by 8. 31Move thedecimal point 3 places to the right (add three zeros).Example:32 The 2-digit number chosen to multiply by 125 is 34. 33Divide by 8:34/8 = 4.25 34 Move the decimal point 3 places to the right:4250 35So 125 x 34 = 4,250.See the pattern?36 The 2-digit number chosen to multiply by 125 is 78. 37Divide by 8:Multiplying a 2-digit number by 1985 Select a 2-digit number. 6 Double the number. 7 If thedouble has2 digits, subtract 1: X X _ _ Subtract the double from 100: _ _XX8 If the double has 3 digits, subtract 2: X X X _ _ (if it has 2zeros, subtract the first digit)Subtract the last 2 digits of the double from 100: _ _ _ X XExample:9 The 2-digit number chosen to multiply by 198 is 21. 10Double it: 2x 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. oubleit: 2 x 56 = 112.Subtract 2: 112 - 2 = 110: 1 1 0 _ _ 100 - doubles 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 a 2- or 3-digit number. S 15 Divide by 4 (or by 2twice). e 16Move the decimal point one place to the right, or add a zero. etExample: he 17 If the 2-digit number chosen to multiply by 2 1/2 is 98 18Divideby 2: 98/2 = 49 ap 19 Divide by 2: 49/2 = 24.5. t20 Move the decimalpoint 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 23Divideby 2: 123/2 = 61.5 24 Divide by 2: 61.5/2 = 30.75. 25 Move thedecimalpoint 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 488Divide by 4: 488/4 = 122 Divide by 2: 61.5/2 = 30.75. Move thedecimalpoint 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 itslast two digits are divisible by 4), or divide by 2 twice. This methodshould becme an easy one.Multiplying a 2-digit number by 2 1/328 Select a 2-digit number. 29 Multiply the number by 7. 30Divide by 3.Example:31 The 2-digit number chosen to multiply by 2 1/3 is 67. 32Multiply by 7:7 x 67 = 420 + 49 = 469. 33 Divide by 3: 469/3 = 156 1/3. 34 So67 x 2 1/3= 156 1/3.
  • 108. See the pattern?35 The 2-digit number chosen to multiply by 2 1/3 is 91. 36Multiply by 7:7 x 91 = 630 + 7 = 637. 37 Divide by 3: 637/3 = 212 1/3. 38 So91 x 2 1/3= 212 1/3.Multiplying a 2-digit number by 2 1/439 Select a 2-digit number. 40 Multiply the number by 9. 41Divide by 4.Example:42 The 2-digit number chosen to multiply by 2 1/4 is 51. 43Multiply by 9:9 x 51 = 450 + 9 = 459. 44 Divide by 4: 459/4 = 114 3/4. 45 So51 x 2 1/4 =114 3/4.See the pattern?46 The 2-digit number chosen to multiply by 2 1/3 is 28. 47Multiply by 9:9 x 28 = 180 + 72 = 252. 48 Divide by 4: 252/4 = 63. 49 So 51 x2 1/4 = 63.Multiplying a 2-digit number by 2 1/550 Select a 2-digit number. 51 Multiply the number by 11. Addthe digits right 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 is473. 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 is 5, carry1. 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 2/359 Select a 2-digit number. 60 Multiply the number by 8. 61Divide by 3.Example:62 The 2-digit number chosen to multiply by 2 2/3 is 24. 63Multiplyby 8: 8 x 24 = 160 + 32 = 192. 64 Divide by 3: 192/3 = 64. 65 So24 x2 2/3 = 64.See the pattern?66 The 2-digit number chosen to multiply by 2 2/3 is 68. 67MultiplyMultiplying a 2-digit number by 2 2/981 A d d o70 Select a 2-digit number. n 71 Multiply by two. e 72 Add onezero.z 73 Divide by nine. er Example: o :74 The 2-digit number chosen to multiply by 2 2/9 is 31. 175 Multiply by 2: 2 x 31 = 62 76 Add one zero: 620 77 Divideby 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 d79 The 2-digit number chosen to multiply by 2 2/9 is 74. 80Multiply by2: 2 x 74 = 148 bey 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 theirults may be repeating decimals. Your steps will produceexact results.Multiplying a 2-digit number by 25084 Select a 2-digit number. 85 Divide the number by 4. 86Move thedecimal point 3 places to the right (add three zeros).Example:87 The 2-digit number chosen to multiply by 250 is 96. 88Divide 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. 92Divide by 4:37/4 = 9.25 93 Move the decimal point 3 places to the right:9250 94 So 37 x 250 = 9,250.Multiplying a 2-digit number by 29795 Select a 2-digit number. 96 Multiply by 3. 97 If the producthas 2digits, subtract 1: X X _ _ Subtract the product from 100: _ _XX98 If the product has 3 digits, add 1 to the first digit: X X X _ _Subtractthis from the product; Subtract the last 2 digits of theproduct from 100:___XXExample:99 The 2-digit number chosen to multiply by 297 is 33. 100Multiply 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-digit number chosen to multiply by 297 is 92.104 Multiply by 3: 3 x 92 = 276. 105 Add 1 to the first digit: 2 +1=3106 Subtract this from the product: 276-3 = 273: 2 7 3 _ _ 107Subtract 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 1/2109 Select a 2-digit number. 110 If number is even, divide by2,then multiply by 7. 111 If number is odd, multiply by 7, thendivide by 2.Example:4 The 2-digit number chosen to multiply by 3 1/2 is 84. 5Divide by 2:84/2 = 42 6 Multiply by 7: 42 x 7 = 280 +14 = 294 7 So 84 x 31/2 = 294.See the pattern?112 The 2-digit number chosen to multiply by 3 1/2 is 63. 113Multiplyby 7: 63 x 7 = 420 + 21 = 441 114 Divide by 2: 441/2 = 220.5115 So63 x 3 1/2 = 220.5.Multiplying a 2-digit number by 3 1/3118 M 116 Select a 2-digit number. o 117 Divide the numberby 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. 120Divideby 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. 124Divide by
  • 112. 3: 29/3 = 9.6 2/3 125 Move the decimal point one place to theright: 96 2/3126 So 29 x 3 1/3 = 96 2/3.If the division does not come out even, divide to one decimalplace andexpress the result as a fraction. Your answer will be a mixednumberwhen you move the decimal point one place to the right.Multiplying a 2-digit number by 3 3/4127 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. 131Multiplyby 30: 24x30 = 600+120 = 720. 132 Divide by 8: 720/8 = 90. 133So 24 x 3 3/4 = 90.See the pattern?134 The 2-digit number chosen to multiply by 3 3/4 is 52. 135Multiply by 30: 52x30 = 1500+60 = 1560. 136 Divide by 8:1560/8 =195. 137 So 52 x 3 3/4 = 195.number by 375138 Select a 2-digit number. 139 Multiply by 3. 140 Divide by8. 141Example:142 The 2-digit number chosen to multiply by 375 is 38. 143Multiplyby 3: 3 x 38 = 90 + 24 = 114 144 Divide by 8: 114/8 = 14 2/814.25 145Move the decimal point 3 places to the right: 14250 146 So375 x 38= 14250.See the pattern?147 The 2-digit number chosen to multiply by 375 is 64. 148Multiply
  • 113. by 3: 3 x 64 = 180 + 12 = 192 149 Divide by 8: 192/8 = 24 150Movethe decimal point 3 places to the right: 24000 151 So 375 x 64= 24000.Multiply left to right to get products, and express theremainder as afraction that iseasily written as a decimal.Multiplying a 2-digit number by 396S 152 Select a 2-digit number. u 153 Multiply by 4. b 154 If theproducthas 2 digits, subtract 1: X X _ _ tr Subtract the product from100: __ X X a c 155 If the product has 3 digits, add 1 to the firstdigit: t X XX _ _ t Subtract this from the product; he last 2 digits of the product from 100: _ _ _ X X156 If the product has two final zeros, subtract the 1st digitfrom theproduct: X X X _ _ The last 2 digits are 0 0 : _ _ _ 0 0Example:157 The 2-digit number chosen to multiply by 396 is 18. 158Multiplyby 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 =3Subtractfrom product: 272 - 3 = 269: 2 6 9 _ _ 100 - products last 2digits:100 - 72 = 28: _ _ 2 8 So 69 x 396 = 26928.
  • 114. 163 The 2-digit number chosen to multiply by 396 is 50. 164Multiplyby 4: 4 x 50 = 200 165 Subtract 1st digit: 200 - 2 = 198: 1 9 8 __ 166Last 2 digits are 0 0: _ _ _ 0 0 167 So 50 x 396 = 19800.Practice multiplying by 4 from left to right. Repeat the firstdigits ofthe answer, then add the final two. Your speed with this largemultiplication problem will be impressive.Multiplying a 2-digit number by 4 1/2168 Select a 2-digit number. 169 Multiply by 9. 170 Divide by2.Example:171 The 2-digit number chosen to multiply by 4 1/2 is 52. 172Multiply by 9: 52 x 9 = 450 +18 = 468 173 Divide by 2: 468/2 =234174 S o5 2 x 4 1 / 2 = 234.See the pattern?175 The 2-digit number chosen to multiply by 4 1/2 is 91. 176Multiplyby 9: 91 x 9 = 810 + 9 = 819 177 Divide by 2: 819/2 = 409 1/2178 So91 x 4 1/2 = 409 1/2.Multiplying a 2-digit number by 495179 Select a 2-digit number . 180 Multiply by 5. 181 If theproduct has 2 digits, subtract 1: X X _ _ Subtract the product from 100:__XX182 If the product has 3 digits, add 1 to the first digit: X X X __ (dontadd 1 if the last two digits are zeros) Subtract this from theproduct:X X X _ _ Subtract the last 2 digits of the product from 100: ___XX
  • 115. Example:183 The 2-digit number chosen to multiply by 495 is 18. 184Multiplyby 5: 5 x 18 = 90. 185 Subtract 1: 90 - 1 = 89: 8 9 _ _ 186 100 –product: 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. 190189Multiply by 5: 5 x 67 = 300 + 35 = 335. 191 Add 1 to the firstdigit:3+1=4Subtract this from the product: 335-4 = 331: 3 3 1 _ _ Subtractthelast two digits of the product from 100: 100 - 35 = 65: _ _ _ 6 5So495 x 67 = 33165.Multiplying a 2-digit number by 5 5/7192 Multiply the number by 4. 193 Add a zero. 194 Divide by7.Example:195 If the 2-digit number chosen to multiply by 5 5/7 is 32:196 Multiply by 4: 4 x 32 = 128 197 Add a zero: 1280 198 Divide by7: 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 Multiplyby 4:4 x 61 = 244 202 Add a zero: 2440 203 Divide by 7: 2440/7 =348 4/7204 So 61 x 5 5/7 = 348 4/7.Numbers divisible by 7 will produce whole-number answers.With
  • 116. other numbers, those using calculators will get long, inexactdecimals.Multiplying a 2-digit number by 5 5/9214 M 205 Multiply the number by 5. u 206 Add a zero. lt 207Divide by 9.i pExample: l y 208 If the 2-digit number chosen to multiply by5 5/9 is 36: b 209 Multiply by 5: 5 x 36 = 150 + 30 = 180 y210 Add a zero: 1800 5 211 Divide by 9: 1800/9 = 200 : 212 So5 5/7 x 36 = 200. 5 x See the pattern? 7 1 213 If the 2-digitnumberchosen to multiply by 5 5/9 is 71: = 355 215 Add a zero: 3550216Divide 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.Withother numbers, those using calculators will geMultiplying a 2-digit number by 594232 S o59 218 Select a 2-digit number . 4 219 Multiply it by 6. x 220 Ifthe productis 2 digits, subtract 1: 9 X X _ _ 2 Then subtract the productfrom 100: =__XX54 221 If the product is 3 digits, add one to the first digit andsubtract the result from the product: , X X X _ _ Then subtract the last twodigits of theproduct from 100: _ _ _ X X 64 Example: 8.222 The 2-digit number chosen to multiply by 594 is 16. 223Multiply by 6:
  • 117. 6 x 16 = 60 + 36 = 96 224 Subtract 1: 96 - 1 = 95: 9 5 _ _ 225100 - 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. 228Multiply by 6: 6 x 92 = 540 + 12 = 552 229 1st digit + 1: 5 + 1 = 6 230Subtract 6 fromproduct: 552 - 6 = 546: 5 4 6 _ _ 231 100 - products last 2digits: 100 - 52 =48: _ _ _ 4 8vide by 4. 235 Divide the resulting quotient by 4. 236 Movethe decimalpointtwo places to the right.Example:237 The 2-digit number chosen to multiply by 6 1/4 is 56. 238Divide by 4:56/4 = 14 239 Divide by 4 again: 14/4 = 3.5 240 Move thedecimal point twoplacesto 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. 243Divide by 4:82/4= 20.5 244 Divide by 4 again: 20.5/4 = 5.125 245 Move thedecimal point twoplaces to the right: 512.5 246 So 82 x 6 1/4 = 512.5.ultiplying a 2-digit number by 6 2/3247 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. 252Multiply by
  • 118. 2: 2 x 63 = 126. 253 Add a zero: 1260. 254 Divide by 3: 1260/3= 420. 255So 63 x 6 2/3 = 420.See the pattern?256 The 2-digit number chosen to multiply by 6 2/3 is 41. 257Multiply by2: 2 x 41 = 82. 258 Add a zero: 820. 259 Divide by 3: 820/3 =273 1/3. 260 So41 x 6 2/3 = 273 1/3.visible by three, the answer will be a whole number. Ifnot, a calculator will get only an approximation, but youranswer will becorrect.Multiplying a 2-digit number by 625261 Select a 2-digit number. 262 Add a zero and divide by 2.263 Divide by8. 264 Move the decimal point 3 places to the right.Example:265 The 2-digit number chosen to multiply by 625 is 38. 266Add zero: 380; divide by 2: 380/2 = 190 267 Divide by 8: 190/8 = 23 6/8 =23.75 268 Movethe 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. 271Add zero: 720;divide by 2: 720/2 = 360 272 Divide by 8: 360/8 = 45 273 Movethe decimalpoint 3 places to the right: 45000 274 So 625 x 72 = 45,000.Multiplying a 2-digit number by 693279 T 275 Select a 2-digit number . h 276 Multiply it by 7. e 277If theproduct is 2 digits, subtract 1: 2 X X _ _ Then subtract theproduct
  • 119. from 100: _ _ X X d i 278 If the product is 3 digits, add one tothe firstdigit and subtract the result from the product: gX X X _ _Thensubtract the last two digits of the product from 100: _ _ _ X Xit nExample: umber chosen to multiply by 693 is 14. 280 Multiply by 7: 7 x14 = 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. 287285Multiply by 7: 7 x 87 = 560 + 49 = 609 286 1st digit + 1: 6 + 1 =7288 Subtract 7 from product: 609 - 7 = 602: 6 0 2 _ _100 - products last 2 digits: 100 - 9 = 91: _ _ _ 9 1 So 87 x 693= 60,291.Multiplying a 2-digit number by 792289 Select a 2-digit number. 290 Multiply it by 8. 291 Add twozeros.292 Subtract the original number from the result.Example:293 The 2-digit number chosen to multiply by 792 is 22. 294Multiplyby 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. 301299
  • 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 toyourself asa cue for the next.Multiplying a 2-digit number by 8 1/3303 Select a 2-digit number. 304 Add two zeros. 305 Divide byfour.306 Divide by three.Example:307 The 2-digit number chosen to multiply by 8 1/3 is 64. 308Addtwo zeros: 6400. 309 Divide by four: 6400/4 = 1600. 310 Dividebythree: 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. 313Addtwo zeros: 3700. 314 Divide by four: 3700/4 = 925. 315 Dividebythree: 925/3 = 308 1/3. 316 So 37 x 8 1/3 = 308 1/3.BMultiplying a 2-digit number by 875317 Select a 2-digit number. 318 Multiply by 7. 319 Divide by8. 320Move the decimal point 3 places to the right.Example:321 The 2-digit number chosen to multiply by 875 is 38. 322Multiplyby 7: 7 x 38 = 210 + 56 = 266 323 Divide by 8: 266/8 = 33 2/8 =33.25
  • 121. 324 Move the decimal point 3 places to the right: 33250 325So 875 x 38 = 33250.See the pattern?326 The 2-digit number chosen to multiply by 875 is 74. 327Multiplyby 7: 7 x 74 = 490 + 28 = 518 328 Divide by 8: 518/8 = 64 6/8 =64.75329 Move the decimal point 3 places to the right: 64750 330So 875x 74 = 64750.Multiplying a 2-digit number by 8 8/9338 D i v i331 Select a 2-digit number. d 332 Multiply by eight. e 333Add onezero. b 334 Divide by nine. y 9Example: :3 335 The 2-digit number chosen to multiply by 8 8/9 is 41. 2336Multiply 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. 341Multiply by 8: 8 x 62 = 480 + 16 = 496 342 Add one zero: 4960343 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 answerswill be exact.Multiplying a 2-digit number by 891345 Select a 2-digit number. 346 Multiply it by 9. 347 If theproducis 2 digits, subtract 1: X X _ _ Then subtract the product from100:__XX
  • 122. 348 If the product is 3 digits, add 1 to the first digit andsubtract itfrom the product: X X X _ _ Then subtract the last 2 digits oftheproduct from 100: _ _ _ X XExample:349 The 2-digit number chosen to multiply by 891 is 11. 350Multiply by 9: 9 x 11 = 99 351 Subtract 1: 99 - 1 = 98: 9 8 _ _352 Subtract the product from 100: 100 - 99 = 1: _ _ 0 1353 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 dd 1 to the first digit: 3 + 1 = 4 356 Subtract 4 from theproduct: 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 999359 Select a 2-digit number. 360 Add one zero to the number,subtract1: X X X _ _361 Subtract the original number from 100: _ _ _ X XExample:362 The 2-digit number chosen to multiply by 999 is 64. 363Add onezero and subtract 1: 640 - 1 = 639: 6 3 9 _ _364 Subtract 64 from 100: 100 - 64 = 36: X X X 3 6365 So 64 x 999 = 63,936.See the pattern?8 The 2-digit number chosen to multiply by 999 is 75. 9 Addone zeroand subtract 1: 750 - 1 = 749: 7 4 9 _ _10 Subtract 75 from 100: 100 - 75 = 25: X X X 2 54. So 64 x 999 = 74,925.
  • 123. Multiplying a 2-digit number by 1001 and adding 21366 Select a 2-digit number. 367 Write it twice. 368 Write one zerobetweenthe 2-digit numbers. 369 Add 21.Example:370 12 x 1001 = 12012 12012 + 21 = 12033 371 So 12 x 1001 +21 = 12033Multiplying a 3-digit number by 1001 and adding 201372 Select a 3-digit number. 373 Write it twice. 374 Add 201.Example:375 123 x 1001 = 123123 376 123123 + 201 = 123324 377 So123 x 1001 +201 = 123324Create variations by changing the number to be added or bygiving anumber to be subtracted. In either case, make the number aneasy oneto handle mentallyMultiplying a 2- or 3-digit number by 16 2/3S 378 Select a 2-digit number. (Choose larger numbers whenyou feelsure about the method.) e 379 Add two zeros to the number. e380 Divideby 6 (or by 2 and 3). thExample:e381 The 2-digit number chosen to multiply by 16 2/3 is 96. ap 382 Add twozeros: 9600. t383 Divide by 3: 9600/3 = 3200. (9600 is divisibleby 3 becausethe sum of its digits is tdivisible by 3.)e384 Divide by 2: 3200/2 = 1600. r 385 So 96 x 16 2/3 = 1600. n
  • 124. ?386 If the 2-digit number chosen to multiply by 16 2/3 is 76:387 Add twozeros: 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 2first. Withsome practice you will become adept at multiplying by 16 2/3Multiplying a 2-digit number by 10001 and adding 21The product of a 2-digit number and 10,001 + 21 is:391 the two digits, 392 two zeros, 393 the two digits, 394 add21.Example: 29 x 10001 + 21395 29 x 10001 = 290029 396 290029 + 21 = 290050Multiplying a 3-digit number by 10001 and adding 201The 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 + 201437 x 10001 = 43704372. 4370437 + 201 = 4370638Multiplying a 4-digit number by 10001 and adding 2001The product of a 4-digit number and 10,001 + 2001 is:the four digits, repeated onceadd 2001.Example: 6741 x 10001 + 20016741 x 10001 = 6741674167416741 + 2001 = 67418742Create variations by changing the number to be added orsubtracted - justmake the number an easy one to handle mentally.Multiplying a 2-, 3-, 4-, or 5-digit numberf by 100001 ir
  • 125. 1. The product of a 2-digit number and 100,001 is: s t first thetwo digits,then three zeros, tthen the two digits again. h eExamples: t h12 x 100001 = 1200012 r79 x 100001 = 7900079 e e2. The product of a 3-digit number and 100,001 is: digits,then two zeros,then the three digits again.Examples:386 x 100001 = 38600386 914 x 100001 = 91400914 3. Theproduct of a4-digit number and 100,001 is:first the four digits,then one zero,then the four digits again.Examples:6295 x 100001 = 6295062958106 x 100001 = 8106081064. The product of a 5-digit number and 100,001 is:first the five digits,then the five digits again.Examples:12345X100001 = 123451234598765X100001 = 9876598765Your friends will need a BIG calculator to keep up with youwhen youannounce these products!Multiplying two 2-digit numbers(difference of 2)400 Choose a 2-digit number. 401 Select a number either 2smaller or
  • 126. 2 larger. 402 Square 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 secondnumber. 405 Theaverage of 29 and 31 is 30. Square 30: 30 x 30 = 900. 406Subtract 1: 900- 1 = 899. 407 So 29 x 31 = 899.See the pattern?408 If the first number is 76, choose 74 as the secondnumber. 409 Theaverage of 76 and 74 is 75. Square 75: 75 x 75 = 5625. 410Subtract 1:5625 - 1 = 5624. 411 So 76 x 74 = 5624.Practice this shortcut (remember your methods of squaringnumbers).Multiplying two 2-digit numbers(difference of 3)412 Choose a 2-digit number. 413 Select a number either 3smaller or 3larger. 414 Add 1 to the smaller number, then square. 415Subtract onefrom the smaller number. 416 Add this to the square.Example:417 If the first number is 27, choose 24 as the secondnumber. 418 Add1 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 thisto 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 secondnumber. 427424
  • 127. Add 1 to the smaller number: 31 + 1 = 32. 425 Square thisnumber: 32 x32 = 1024. 426 Subtract one from the smaller number: 31 - 1 =30. 428Add this to the square: 1024 + 30 = 1054.So 34 x 31 = 1054.Choose the second number that will give you the easiersquare, and useyour square shortcuts.Multiplying two 2-digit numbers(difference of 4)429 Choose a 2-digit number. 430 Select a number either 4smaller or 4larger. 431 Find the middle number of the two (the average).432 Squarethis middle number. 433 Subtract 4 from this square.Example:434 If the first number is 63, choose 67 as the secondnumber. 435 Themiddle number (the average) is 65. 436 Square this middlenumber: 65 x65 = 4225. (Remember how to square a 2-digit numberending in 5?) 437 Subtract 4 from this square: 4225 - 4 =4221. 438 So 63x 67 = 4221.See the pattern?439 If the first number is 38, choose 42 as the secondnumber. 440 Themiddle number (the average) is 40. 441 Square this middlenumber: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 easiersquare, and use
  • 128. your square shortcuts.Multiplying two 2-digit numbers443 C (difference of 6) hoose a 2-digit number. 444 Select a number either 6 smaller or6 larger.445 Find the middle number of the two (the average). 446Square thismiddle number. 447 Subtract 9 from this square.Example:448 If the first number is 78, choose 72 as the secondnumber. 449The middle number (the average) is 75. 450 Square thismiddle number: 75 x 75 = 5625. (Remember how to square a 2-digit numberending in 5?) 451 Subtract 9 from this square: 5625 - 9 =5616. 452 So78 x 72 = 5616.See the pattern?453 If the first number is 31, choose 37 as the secondnumber. 456454The middle number (the average) is 34. 455 Square thismiddle number: 34 x 34 = 1156. (Remember how to square a 2-digit numberending 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 easiersquare, and useyour square shortcuts.Practice!Multiplying two 2-digit numbers13 S (difference of 8) qu 458 Choose a 2-digit number. a 459 Select a number either8 smaller or
  • 129. 8 larger. r 460 Find the middle number of the two (theaverage). e 461Square this middle number (multiply it by itself). t 462Subtract 16 fromthis square. hi Example: sm 11 If the first number is 34, choose 26 as the secondnumber (8 smaller). i 12 The middle number (the average) is 30. ddle 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 secondnumber (8 larger).464 The middle number (the average) is 68. 465 Square thismiddlenumber: 68 x 68 = 4624. (Remember how to square a 2-digitnumberending 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 easiersquare, anduse your square shortcuts.Practice!Multiplying two 2-digit numberse (difference of 10) nd 467 Choose a 2-digit number. i 468 Select a number either10 smalleror 10 larger. n 469 Find the middle number of the two (theaverage). g470 Square this middle number (multiply it by itself). i 471Subtract 25from this square. n 3 Example: ? )
  • 130. 472 If the first number is 36, choose 26 as the secondnumber (10 smaller). 479 S 473 The middle number (the average) is 31. u 474 Squarethis middlenumber: 31 x 31 = 961. (Remember how to square a 2-digitnumber bending in 1?) tr 475 Subtract 25 from this square: 961 - 25 =936 a(subtract mentally in steps: think 961 - 20 - 5 = 941 - 5 = 936).ct 476 So 36 x 26 = 936. 2 5 See the pattern? f r477 If the first number is 78, you might pick 88 as the secondnumber (10 larger). o 478 The middle number (the average) is 83. m3. Square this middle number: 83 x 83 = 6889. (Rememberhow to squarea 2-digit number t his square: 6889 - 25 = 6864 (subtractmentally 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 toget an easysquare. With practice you will be a whiz at getting theseproducts.Squaring numbers made up of nines481 Choose a a number made up of nines (up to nine digits).482 Theanswer will have one less 9 than the number, one 8, the samenumberof zeros as 9s, and a final 1Example:483 If the number to be squared is 9999 484 The square ofthe number has:one less nine than the number 9 9 9 one 8 8 the same numberof zeros as 9s 0 0 0 a final 1 1
  • 131. 3. So 9999 x 9999 = 99980001. See the pattern? 485 If thenumber to besquared is 999999 486 The square of the number has:one less nine than the number 9 9 9 9 9one 8 8 3 the same number of zeros as 9s 0 0 0 0 0.a final 1 1So 999999 x 999999 = 999998000001. This is not a verydemanding mentalmath exercise,but it is an interesting pattern.Squaring numbers made up of ones487 Choose a a number made up of ones (up to nine digits).488 Theanswer will be a series of consecutive digits beginning with1, up tothe number of ones in the given number, and back to 1.Example:489 If the number is 11111, (5 digits) -490 The square of thenumber is23454321. (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 thenumberis 1234567654321. (Begin with 1, up to 7, then back to 1.).This is an easy one, but it should be good for a quickexample of yourmental math abilities. Challenge a friend and BEAT THECALCULATOR!Reversing/adding/subtracting 3-digit numbers493 Select a number with different 1st and 3rd digits. 494Reverse thedigits. 495 Subtract the smaller digit from the larger. 496Reverse the
  • 132. digits of this answer. 497 Add these two numbers. 498Subtract anumber from this sum. 499 The final answer will be 1089minuswhatever 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 Theanswer is: 1089 - 22 = 1067.See the pattern?502 For the last step you say to subtract 101. 503 The answeris:1089 - 101 = 988.Each time, change the number you select for the last step –justselect one that is easy to subtract from 1089. Then youranswerswill be immediate and correct.Squaring numbers made up of sixes504 Choose a a number made up of sixes.2. The square is made up of:a. one fewer 4 than there are repeating 6sb. 3c. same number of 5s as 4sd. 6Example:505 If the number to be squared is 666 506 The square of thenumber has:4s (one less than digitsin number) 4 4 3 3 5s (same number as 4s) 5 5 6 63. So 666 x 3666333 = 443556.See the pattern?507 If the number to be squared is 66666The square of the number has:4s (one less than digits
  • 133. in number) 4 4 4 4 3 3 5s (same number as 4s) 5 5 5 5 663. So 66666 x 66666 = 4444355556.Squaring a 2-digit number beginning with 1508 Take a 2-digit number beginning with 1. 509 Square theseconddigit (keep the carry) _ _ X510 Multiply the second digit by 2 and add the carry (keep thecarry)_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 513Multiply 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 _ _ 1518 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 numbersending in 1519 Select a 2-digit number and square it. 520 Drop the lasttwo digitsof the square. 521 Find the largest square root of theremaining digits.This is first digit of the square root.517522 T h e s e c o n d digit is 1.Examples:
  • 134. 523 If the square is 2601:524 Drop the last two digits: 26 525Find thelargest root in 26: 5 x 5 = 25 526 The first digit is 5. Thesecond digitis 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 530Find theargest root in 82: 9 x 9 = 81 531 The first digit is 9. Thesecond digit is 1.532 So the square root of 2601 is 91. 533 534 This processalso worksfor squares of 3-digit numbers (or more).For the square 22801, find the largest root of 228: 15x15 =225. Thefirst two digits are 15, the last digit is 1, and the square rootof 22801 is 151.Finding the square root of perfect squares548 S ending in 5 ot 535 Select a 2-digit number and square it. h 536 Drop thelast twodigits of the square. e 537 Find the largest square root of theremainingdigits. s This is first digit of the square root. qu 538 The second digit is 5. a r Examples: e r 539 If thesquare is 9025: o540 Drop the last two digits: 90 o 541 Find the largest root in90: 9 x 9 =81 t 542 The first digit is 9. The second digit is 5. o 543 So thesquareroot of 9025 is 95. f4 See the pattern? 2 2544 If the square is 4225: 5 545 Drop the last two digits: 42 i546 Find
  • 135. the largest root in 42: 6 x 6 = 36 s 547 The first digit is 6. Thesecond digit is5. 65.This process also works for squares of 3-digit numbers (ormore).For the square 15625, find the largest root of 156: 12x12 =144 (not13x13 = 169). The first two digits are 12, the last digit is 5,and thesquare root of 15625 is 125.Squaring special numbers (1 and repeating 3s)549 Choose a number with a 1 and repeating 3s.2. The square is made up of: first digits: 1 & one fewer 7 thanrepeating 3s next digits: 6 & one fewer 8 than repeating 3slast digit: 9Example:550 If the number to be squared is 13333: 551 The squarehas:first digits: 1 and one fewer 7 than 3s 1 7 7next digits: 6 and one fewer 8 than 3s 68 8 last digit: 9 93. So 1333 x 1333 = 1776889.See the pattern?552 If the number to be squared is 133333:The square has:3first digits: 1 and one fewer .S7 than 3s 1 7 7 7 7onext digits: 6 and one fewer 1 3 8 than 3s 6 8 8 8 8 3 3 lastdigit: 9 9 33 x 133333 = 17777688889.
  • 136. Squaring special numbers (1 and repeating 6s)553 Choose a number with a 1 and repeating 6s.2. The square is made up of: first digits: 2 & one fewer 7 thanrepeating6s next digits: same number of 5s as repeating 6s last digit:6Example:554 If the number to be squared is 1666: 555 The square has:first digits: 2 and one fewer 7 than 6s 2 7 7next digits: same number of 5s as 6s 5 55 last digit: 6 63. 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 6s 27 7 7 7 next digits: same number of 5sas 6s 5 5 5 5 5 last digit: 6 63. So 166666 x 166666 = 27777555556.Squaring special numbers (1 and repeating 9s)n 557 Choose a number with a 1 and repeating 9s. e2. The square is made up of: x first digits: 3 & one fewer 9than repeating 9stdigits: 6 & one fewer 0 than repeating 9s last digit: 1Example:558 If the number to be squared is 1999: 559 The square has:first digits: 3 and one fewer 9 than 9s 3 9 9next digits: 6 and one fewer 0 than 9s 60 0 last digit: 1 13. 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 9s 3
  • 137. 9 9 9 9 next digits: 6 and one fewer 0than 9s & 6 0 0 0 0 last digit: 1 13. So 199999 x 199999 = 39999600001.Squaring special numbers (2 and repeating 3s)561 Choose a number with a 2 and repeating 3s.2. The square is made up of: first digits: 5 & one fewer 4 thanrepeating3s next digits: 2 & one fewer 8 than repeating 3s last digit: 9Example:562 If the number to be squared is 2333: 563 The square has:first digits: 5 and one fewer 4 than 3s 5 4 4next digits: 2 and one fewer 8 than 3s 28 8 last digit: 9 93. So 2333 x 2333 = 5442889.564 If the number to be squared is 233333:The square has:first digits: 5 and one fewer4 than 3s 5 4 4 4 4next digits: 2 and one fewer8 than 3s 2 8 8 8 8last digit: 9 93. So 233333 x 233333 = 54444288889.Squaring special numbers (2 and repeating 6s)565 Choose a number with a 2 and repeating 6s.2. The square is made up of:first digits: 7 & two fewer 1s than repeating 6snext digits: 07 & one fewer 5 than repeating 6slast digit: 6Example:566 If the number to be squared is 2666: 567 The square has:first digits: 7 and two fewer 1s than 6s 7 1next digits: 07 and one fewer 5 than 6s 07 5 5 last digit: 6 63. So 2666 x 2666 = 7,107,556.See the pattern?
  • 138. 568 If the number to be squared is 266666:The square has:first digits: 7 and two fewers than 6s 7 1 1 1next digits: 07 and one fewer5 than 6s 0 7 5 5 5 5last digit: 6 63. So 266666 x 266666 = 71,110,755,556.Squaring special numbers (2 and repeating 9s)569 Choose a number with a 2 and repeating 9s.2. The square is made up of: first digits: 8 & one fewer 9 thanrepeating9s next digits: 4 & one fewer 0 than repeating 9s last digit: 1Example:570 If the number to be squared is 2999: 571 The square has:first digits: 8 and one fewer 9 than 9s 8 9 9next digits: 4 and one fewer 0 than 9s 40 0 last digit: 1 13. So 2999 x 2999 = 8,994,001.See the pattern?9 If the number to be squared is 299999:The square has:3first digits: 8 and one fewer .S9 than 9s 8 9 9 9 9onext digits: 4 and one fewer 2 9 0 than 9s 4 0 0 0 0 9 9 lastdigit: 1 1 99 x 299999 = 89,999,400,001.Squaring special numbers (3 and repeating 6s)Choose a number with a 3 and repeating 6s.2. The square is made up of: first digits: 13 & two fewer 4sthan repe
  • 139. ating 6s next digits: 39 & one fewer 5 than repeating 6s lastdigit: 6Example:If the number to be squared is 3666: The square has:first digits: 13 and two fewer 4s than 6s 1 34 next digits: 39 and one fewer 5 than 6s3 9 5 5 last digit: 6 63. 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 fewer4s than 6s 1 3 4 4 4next digits: 39 and one fewer5 than 6s 3 9 5 5 5 5last digit: 6 63. So 366666 x 366666 = 134,443,955,556.Squaring special numbers (3 and repeating 9s)Choose a number with a 3 and repeating 9s. .The square is made up of: first digits: 15 & one fewer 9 thanrepeating9s next digits: 2 & one fewer 0 than repeating 9s last digit: 1Example:If the number to be squared is 3999: The square has:first digits: 15 and one fewer 9 than 9s 1 5 99 next digits: 2 and one fewer 0 than 9s 20 0 last digit: 1 13. 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 fewer9 than 9s 1 5 9 9 9 9next digits: 2 and one fewer0 than 9s 2 0 0 0 0
  • 140. last digit: 1 13. So 399999 x 399999 = 159,999,200,001.Squaring special numbers (4 and repeating 3s)Choose a number with a 4 and repeating 3s. E2. The square is made up of: first digits: 18 & one fewer 7than repeating3s a next digits: 4 & one fewer 8 than repeating 3s m lastdigit: 9 ple:If the number to be squared is 4333: The square has:first digits: 18 and one fewer 7 than 3s 1 8 77 next digits: 4 and one fewer 8 than 3s 48 8 last digit: 9 93. 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 fewer7 than 3s 1 8 7 7 7 7next digits: 4 and one fewer8 than 3s 4 8 8 8 8last digit: 9 93. So 433333 x 433333 = 187,777,488,889.Squaring special numbers (4 and repeating 6s)Choose a number with a 4 and repeating 6s.2. The square is made up of: first digits: 21 & one fewer 7thanrepeating 6s next digits: 1 & one fewer 5 than repeating 6slast digit: 6Example:If the number to be squared is 4666: The square has:first digits: 21 and one fewer 7 than 6s 2 1 77 next digits: 1 and one fewer 5 than 6s 1:663. So 4666 x 4666 = 21,771,556.
  • 141. See the pattern?If the number to be squared is 466666:The square has:first digits: 21 and one fewer7 than 6s 2 1 7 7 7 7next digits: 1 and one fewer5 than 6s 1 5 5 5 5last digit: 6 63. So 466666 x 466666 = 217,777,155,556.Squaring special numbers (4 and repeating 9s)S Choose a number with a 4 and repeating 9s. e2. The square is made up of: e first digits: 24 & one fewer 9thanRepeating 9s in the number tnext digits: same numberof 0s as repeating 9s in the number hlast digit: 1eExample: patIf the number to be squared is 4999: tThe square has:erfirst digits: 24 and one fewer 9 than 9s (innthe number) 2 4 9 9 next digits: same number ? of 0s as 9s(in thenumber) 0 0 0 last digit: 13. So 4999 x 4999 = 24,990,001.If the number to be squared is 499999:The square has:first digits: 24 and onefewer 9 than 9s 2 4 9 9 9 9next digits: same numberof 0s as 9s 0 0 0 0 0last digit: 1 13. So 499999 x 499999 = 249,999,000,001.Squaring special numbers (5 and repeating 3s)
  • 142. Choose a number with a 5 and repeating 3s.2. The square is made up of: first digits: 28 & one fewer 4than repeating3s next digits: 0 & one fewer 8 than repeating 3s last digit: 9Example:If the number to be squared is 5333: The square has:first digits: 28 and one fewer 4 than 3s 2 8 44 next digits: 0 and one fewer 8 than 3s 08 8 last digit: 9 93. So 5333 x 5333 = 28,440,889.See the pattern? If the number to be squared is 533333: Thesquare has:first digits: 28 and one fewer4 than 3s 2 8 4 4 4 4next digits: 0 and one fewer3s 0 8 8 8 8last digit: 9 93. So 533333 x 533333 = 284,444,088,889.Squaring special numbers (5 and repeating 6s)Choose a number with a 5 and repeating 6s.2. The square is made up of: first digits: 32 & two fewer 1sthanrepeating 6s next digits: 03 & one fewer 5 than repeating 6slast digit: 6Example:If the number to be squared is 5666: The square has:first digits: 32 and two fewer 1s than 6s 3 21 next digits: 03 and one fewer 5 than 6s0 3 5 5 last digit: 6 63. So 5666 x 5666 = 32,103,556.See the pattern? If the number to be squared is 566666: Thesquare has:first digits: 32 and two fewer1s than 6s 3 2 1 1 1next digits: 03 and one fewer
  • 143. 1 than 6s 0 3 5 5 5 5last digit: 6 63. So 566666 x 566666 = 321,110,355,556.Squaring special numbers (5 and repeating 9s)Choose a number with a 5 and repeating 9s.2. The square is made up of: first digits: 35 & two fewer 9sthanrepeating 9s next digits: 88 & one fewer 0 than repeating 9slast digit: 1Example:If the number to be squared is 5999: The square has:first digits: 35 and two fewer 9s than 9s 3 59 next digits: 88 and one fewer 0 than 9s8 8 0 0 last digit: 1 13. 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 fewer9s than 9s 3 5 9 9 9next digits: 88 and one fewer0 than 9s 8 8 0 0 0 0last digit: 1 13. So 599999 x 599999 = 359,998,800,001.Squaring special numbers (6 and repeating 3s)Choose a number with a 6 and repeating 3s. f2. The square is made up of: t first digits: 40 & two fewer 1sthanrepeating 3s h next digits: 06 & one fewer 8 than repeating3s e last digit: 9nuExample:mber to be squared is 6333: The square has:first digits: 40 and two fewer 1s than 3s 4 0
  • 144. 1 next digits: 06 and one fewer 8 than 3s0 6 8 8 last digit: 9 93. 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 fewer1s than 3s 4 0 1 1 1next digits: 06 and one fewer8 than 3s 0 6 8 8 8 8last digit: 9 93. So 633333 x 633333 = 401,110,688,889.Squaring special numbers (6 and repeating 9s)S Choose a number with a 6 and repeating 9s. e2. The square is made up of: e first digits: 48 & two fewer 9sthanrepeating 9s in the number tnext digits: 86 & one fewer 0thanrepeating 9s in the number hlast digit: 1eExample: patIf the number to be squared is 6999: tThe square has:erfirst digits: 48 and two fewer 9s than rep.n9s 4 8 9 next digits: 86 and one fewer 0 ? than rep. 9s 8 6 0 0last digit: 1 13. So 6999 x 6999 = 48,986,001. fthe number to be squared is 699999:The square has:first digits: 48 and two fewer9s than rep. 9s 4 8 9 9 9next digits: 86 and one fewer0 than rep. 9s 8 6 0 0 0 0
  • 145. last digit: 1 13. So 699999 x 699999 = 489,998,600,001.Squaring special numbers (7 and repeating 3s)Choose a number with a 7 and repeating 3s.2. The square is made up of: first digits: 53 & one fewer 7thanrepeating 3s next digits: 2 & one fewer 8 than repeating 3slast digit: 9Example:If the number to be squared is 7333: The square has:first digits: 53 and one fewer 7 than 3s 5 3 77 next digits: 2 and one fewer 8 than 3s 28 8 last digit: 9 93. So 7333 x 7333 = 53,772,889.See the pattern? If the number to be squared is 733333: Thesquare has:igit: 93. So 733333 x733333 =537,777,288,889.Squaring special numbers (7 and repeating 6s)Choose a number with a 7 and repeating 6s.2. The square is made up of: first digits: 58 & two fewer 7sthanrepeating 6s next digits: 67 & one fewer 5 than repeating 6slast digit: 6Example:If the number to be squared is 7666: The square has:first digits: 58 and two fewer 7s than 6s 5 87 next digits: 67 and one fewer 5 than 6s6 7 5 5 last digit: 6 63. So 7666 x 7666 = 58,767,556.See the pattern?first digits: 53 and one fewer7 than 3s 5 3 7 7 7 7
  • 146. next digits: 2 and one fewer8 than 3s 2 8 8 8 8If the number to be squared is 766666:The square has:first digits: 58 and two fewerthan 6s 5 8 7 7 7next digits: 67 and one fewer5 than 6s 6 7 5 5 5 5last digit: 6 63. So 766666 x 766666 = 587,776,755,556.Squaring special numbers (7 and repeating 9s)Choose a number with a 7 and repeating 9s (use a mimimumof three 9s).2. The square is made up of: first digits: 63 & two fewer 9sthan repeating 9s in the number next digits: 84 & one fewer 0 than repeating 9s in the number last digit: 1Example:If the number to be squared is 7999: The square has:first digits: 63 and two fewer 9s than rep.9s 6 3 9 next digits: 84 and one fewer 0than rep. 9s 8 4 0 0 last digit: 1 13. So 7999 x 7999 = 63,984,001.See the pattern?If the number to be squared is 799999:The square has:3first digits: 63 and two fewer .S9s than rep. 9s 6 3 9 9 9onext digits: 84 and one fewer 7 9 0 than rep. 9s 8 4 0 0 0 0 99 last digit: 1 1 9 9x 799999 = 639,998,400,001.Squaring special numbers (8 and repeating 3s)Choose a number with an 8 and (at least 3) repeating 3s.
  • 147. 2. The square is made up of: first digits: 69 & two fewer 4sthan repeating 3s next digits: 3 & same number of 8s asrepeating 3s last digit: 9Example:If the number to be squared is 8333: The square has:first digits: 60 and two fewer 4s than 3s 6 94 next digits: 3 and same number of 8s as3s 3 8 8 8 last digit: 9 93. So 8333 x 8333 = 69,438,889.See the pattern? If the number to be squared is 833333: Thesquare has:first digits: 69 and two fewer4s than 3s 6 9 4 4 4next digits: 3 and same number of8s as 3s 3 8 8 8 8 8last digit: 9 93. So 833333 x 833333 = 694,443,888,889.Squaring special numbers (8 and repeating 9s)Choose a number with an 8 and repeating 9s (use amimimumof three 9s). a2. The square is made up of: s first digits: 80 & two fewer 9sthan repeating 9s in the number t next digits: 82 & one fewer 0than repeating 9s in the number d igit: 1Example:If the number to be squared is 8999: The square has:first digits: 80 and two fewer 9s than rep.9s 8 0 9 next digits: 82 and one fewer0 than rep. 9s 8 2 0 0 last digit: 1 13. 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. 9s than rep. 9s 8 0 9 9 9next digits: 82 and one fewer0 than rep. 9s 8 2 0 0 0 0last digit: 1 13. So 899999 x 899999 = 809,998,200,001.Squaring special numbers (9 and repeating 3s)Choose a number with a 9 and repeating 3s (use at leastthree 3s).2. The square is made up of: first digits: 87 & two fewer 1sthanrepeating 3s next digits: 04 & one fewer 8 than repeating 3slast digit: 9Example:If the number to be squared is 9333: The square has:: 87 and two fewer 1s than 3s 8 7 1 nextdigits: 04 and one fewer 8 than 4s 0 4 8 8last digit: 9 93. So 9333 x 9333 = 87,104,889.See the pattern? If the number to be squared is 933333: ihesquare has:first digits: 87 and two fewer1s than 3s 8 7 1 1 1next digits: 04 and one fewer8 than 3s 0 4 8 8 8 8last digit: 9 93. So 933333 x 933333 = 871,110,488,889.Squaring special numbers (9 and repeating 6s)Choose a number with a 9 and repeating 6s (use at leastthree 6s).2. The square is made up of: first digits: 93 & two fewer 4sthanrepeating 6s next digits: 31 & one fewer 5 than repeating 6slast digit: 6Example:If the number to be squared is 9666: The square has:
  • 149. first digits: 93 and two fewer 4s than 6s 9 34 next digits: 31 and one fewer 5 than 6s3 1 5 5 last digit: 6 63. So 9666 x 9666 = 93,431,556.See the pattern? If the number to be squared is 966666: Thesquare has:first digits: 93 and two fewer4s than 6s 9 3 4 4 4next digits: 31 and one fewer5 than 6s 3 1 5 5 5 5last digit: 6 63. So 966666 x 966666 = 934,443,155,556.Squaring numbers in the 20sSquare the last digit (keep the carry) _ _ X Multiply the lastdigitby 4, add the carry _ X _ The first digit will be 4 plus thecarry: X _ _Example:If the number to be squared is 24:Square the last digit (keep the carry):4 x 4 = 16 (keep 1) _ _ 6Multiply 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) _ _ 6Multiply 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 last digit (keep the carry) _ _ _ X Multiply the lastdigit by 6, add the carry _ _ X _ The first digits will be 9 plusthe 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) _ _ _6Multiply 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 = 1111__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) _ _ _6Multiply 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 = 121 2 _ _.So 36 x 36 = 1296.Squaring numbers in the 40sSquare the last digit (keep the carry) _ _ X Multiply the lastdigitby 8, add the carry _ X _ f The first digits will be 16 plus thecarry: X X _ _t Example: he number to be squared is 42:Square the last digit: 2 x 2 = 4 _ _ _ 4Multiply 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?If the number to be squared is 48:Square the last digit (keep the carry):8 x 8 = 64 (keep 6) _ _ _ 4Multiply 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 50s2 Square the last digit (keep the carry) _ _ _ X 5 Multiply thelastdigit by 10, add the carry _ _ X _ (The first digits will be 25plus the carry: X X _ _ + c Example:a r If the number to be squaredis53: r y Square the last digit (keep the carry): ): 3 x 3 = 9 (keep3)_ _ _ 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: 28 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) _ _ _6Multiply 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 60sSquare the last digit (keep the carry) _ _ _ X Multiply the lastdigitby 12, add the carry _ _ X _ The first digits will be 36 plus thecarry: X X _ _
  • 152. Example:If the number to be squared is 63:Square the last digit (keep the carry):3 x 3 = 9 (keep 3) _ _ _ 9Multiply 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 (keepthecarry): 7 x 7 = 49 (keep 4) _ _ _ 9Multiply 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 70sSquare the last digit (keep the carry) _ _ _ X Multiply the lastdigit by 14, add the carry _ _ X _ The first digits will be 49plus the carry: X X__Example:If the number to be squared is 72:Square the last digit: 2 x 2 = 4 _ _ _ 4Multiply 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 last digit by 14, add the carry:14 x 8 = 80 + 32 = 112112 + 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 80sSquare the last digit (keep the carry) _ _ X Multiply the lastdigit by16, add the carry _ X _ The first digits will be 64 plus thecarry: X X _ _Example:If the number to be squared is 83:Square the last digit: 3 x 3 = 9 _ _ _ 9Multiply 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) _ _ _6Multiply 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 200sThenChoose a number in the 200s (practice with numbers under210, then progress to larger ones). e The first digit of the squareis 4: 4 _ _ _ _
  • 154. t two digits will be 4 times the last 2 digits: _ X X _ _ The lasttwoplaces will be the square of the last digit: _ _ _ X XExample: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: _ 24__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 timesthelast 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(keepcarry): 25x25 = 625 (keep 6): _ _ _ 2 5 4 times the last twodigits+ carry: 4x25 = 100; 100+6 = 106 (keep 1): _ 0 6 _ _ Square thefirst digit + carry: 2x2 = 4; 4+1 = 5: 5 _ _ _ _ So 225 x 225 =50625.Squaring numbers in the 300s Sq Choose a number in the 300s (practice with numbers under310, then progress to larger ones). u The first digit of thesquareis 9: 9 _ _ _ _ a The next two digits will be 6 times the last 2digits:_ X X _ _ r The last two places will be the square of the lastdigit: _ _ _ X X et Example: h eIf the number to be squared is 309: The first digit is 9: 9 _ _ __aThe next two digits are 6 times the last digit: s 6 x 9 = 54: _ 5 4
  • 155. _ _ t digit: 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 timesthelast 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 last two digits + carry: 6x25 = 150; 150+6 = 156 (keep 1): _ 5 6 __Square the first digit + carry: 3x3 = 9; 9+1 = 10: 1 0 _ _ _ _ So325 x 325 =105625.Squaring numbers in the 400sS Choose a number in the 400s (keep the numbers low atfirst;then progress to larger ones). qThe first two digits of thesquareare 16: 1 6 _ _ _ _ u The next two digits will be 8 times the last2digits: _ _ X X _ _ a The last two places will be the square ofthelast two digits: _ _ _ _ X X re Example: th If the number to be squared is 407: e The first two digits are16: 1 6 __ _ _ l The next two digits are 8 times the last 2 digits: a 8 x 7= 56: _ _ 5 6 __st Square the last digit: 7 x 7 = 49: _ _ _ 4 9 t So 407 x 407 =165,649.wo For larger numbers reverse the steps: di
  • 156. Square the last two digits (keep the carry): _ _ _ _ X X g 8times the last two digits + carry: _ _ X X _ _ it 16 + carry: X X _ _ _ _ s(kSee the pattern?eIf the number to be squared is 425: pethe carry): 25 x 25 = 625 (keep 6): _ _ _ _ 2 58 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 500sChoose a number in the 500s (start with low numbers at first;then graduate to larger ones). The first two digits of thesquareare 25: 2 5 _ _ _ _ The next two digits will be 10 times the last2digits: _ _ X X _ _ The last two places will be the square ofthelast two digits: _ _ _ _ X XExample: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 10times thelast two digits + carry: _ _ X X _ _25 + carry: X X _ _ _ _
  • 157. If the number to be squared is 525: Square the last two digits(keepthe carry): 25 x 25 = 625 (keep 6): _ _ _ _ 2 5 10 times the lasttwo digits + carry: 10 x 25 = 250; 250 + 6 = 256 (keep 2): _ _ 5 6 _ _25 + carry:25 + 2 - 27: 2 7 _ _ _ _Squaring numbers in the 600sChoose a number in the 600s (practice with smaller numbers,thenprogress to larger ones). S The first two digits of the squareare 36: 3 6 _ _ _ _ qThe next two digits will be 12 times the last 2digits: __ X X _ _ u The last two places will be the square of the lasttwo digits:____XXar Example: e t If the number to be squared is 607: h The firsttwo digitsare 36: 3 6 _ _ _ _ e The next two digits are 12 times the last 2digits:12 x 07 = 84: _ _ 8 4 _ _ ast 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): _ _ _ _ 2 512 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 700s3 Choose a number in the 700s (practice with smallernumbers, then
  • 158. progress to larger ones). . Square the last two digits (keep thecarry):_ _ _ _ X X Multiply the last two digits by 14 and M add thecarry: _ _ XX__ult The first two digits will be 49 plus the carry: X X _ _ _ _ ipExample: ly If the number to be squared is 704: t Square the last twodigits (keep the carry): h4 x 4 = 16: _ _ _ _ 1 6 el Multiply the last two digits by 14 and a add the carry: 14 x 4= 56: _ _ 56__st The first two digits will be 49 plus the carry: 4 9 _ _ _ _ t So704 x 704 = 495,616. wo See the pattern? d iIf the number to be squared is 725: g Square the last twodigits (keep the carry):it 25 x 25 = 625: _ _ _ _ 2 5 sb y 14 and add the carry: 14 x 25 = 10 x 25 + 4 x 25 = 250 +100 = 350. 350+6=356: 56: _ _ 5 6 _ _The first two digits will be 49 plus the carry: 49 + 3 = 52: 5 2 __ _ _ So725 x 725= 525,625.Squaring numbers between 800 and 810Choose a number between 800 and 810. Square the last twodigits: ____XXMultiply 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 twodigits: 2 x 2 =4: _ _ _ _ 0 4Multiply 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 twodigits: 7 x 7 =49: _ _ _ _ 4 9Multiply 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 900se Choose a number in the 900s - start out easy with numbersnear 1000; then go lower whenpert. Subtract the number from 1000 to get the difference.The firstthree places will be the number minus the difference: X X X __ _.The last three places will be the square of the difference: _ __XXX (if 4 digits, add the first digit as carry).Example:If the number to be squared is 985: Subtract 1000 - 985 = 15(difference) Number - difference: 985 - 15 = 970: 9 7 0 _ _ _ Squarethedifference: 15 x 15 = 225: _ _ _ 2 2 5 So 985 x 985 = 970225.See the pattern?
  • 160. If the number to 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 firstdigit when four digits: 8 4 6 _ _ _6. So 920 x 920 = 846400.Squaring numbers between 1000 and 1100Choose a number between 1000 and 1100. S The first twodigitsare: 1,0 _ _, _ _ _ qFind the difference between your numberand1000. u Multiply the difference by 2: 1,0 X X, _ _ _ a Square thedifference: 1,0 _ _, X X X re Example: th If the number to be squared is 1007: e The first two digitsare: 1,0 __, _ _ _ d Find the difference: 1007 - 1000 = 7 if Two times thedifference: 2 x 7 = 14: f1,0 1 4, _ _ _ erence: 7 x 7 = 49: 1,0 1 4, 0 4 9 So 1007 x 1007 = 1,014,049.Seethe pattern?If the number to be squared is 1012: The first two digits are:1,0_ _, _ _ _ Find the difference: 1012 - 1000 = 12 Two times thedifference: 2 x 12 = 24: 1,0 2 4, _ _ _Square the difference: 12 x 12 = 144: 1,0 2 4, 1 4 4So 1012 x 1012 = 1,024,144.Start with lower numbers and then extend your expertise toallthe numbers between 1000 and 1100. Remember to add thefirstdigit as carry when the square of the difference is four digits.Squaring numbers between 2000 and 2099For
  • 161. Choose a number between 2000 and 2099. (Start withnumbersbelow 2025 to begin with, then graduate to larger numbers.) lThe first two digits are: 4 0 _ _ _ _ _ a The next two digits are4times the last two digits: r 4 0 X X _ _ _ ge For the last three digits, square the last two digits in thenumberchosen (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 digitsare:4 0 _ _ _ _ _ e The next two digits are 4 times the last two: r 4x3=12: _ _ 1 2 _ _ _ s ,For the last three digits, square the lasttwo:3 x 3 = 9: _ _ _ _ 0 0 9 r eSo 2003 x 2003 = 4,012,009. veSee the pattern? rse 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 (keepthecarry): 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 and3099Ifth
  • 162. Choose a number between 3000 and 3099. (Start withnumbersbelow 3025 to begin with, then e graduate to larger numbers.)n The first two digits are: 9 0 _ _ _ _ _ u The next two digitsare6 times the last two digits: m 9 0 X X _ _ _ be For the last three digits, square the last two digits in thenumberchosen (insert zeros when r needed): 9 0 _ _ X X X to Example: beIf 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: diSo 3004 x 3004 = 9,024,016. s3See the pattern? 025For larger numbers, reverse the order: :For the last three digits, square the last two: T 25 x 25 = 625:____625he The middle two digits are 6 times the last two (keep thecarry):6 x 25 = 150 (keep carry of 1): _ _ 5 0 _ fi _ _ r st two digits are90 + the carry: 90 + 1 = 91: 9 1 _ _ _ _ _So 3025 x 3025 = 9,150,625.Squaring numbers between 4000 and 4099fth
  • 163. Choose a number between 4000 and 4099. e For numberslessthan 4013: n The first three digits are: 1 6 0 _ _ _ _ _ u Thenext two digits are 8 times the last two digits: m _ _ _ X X _ _ _ be For the last three digits, square the last two digits in thenumber chosen (insert zeros when r needed): _ _ _ _ _ X X X to Example: beIf the number to be squared is 4005: s The first three digitsare:1 6 0 _ _ _ _ _ q The next two digits are 8 times the last two: u8x5 = 40: _ _ 4 0 _ _ _ arFor the last three digits, square the last two: e 5 x 5 = 25: _ _ ___025diSo 4005 x 4005 = 16,040,025. s4See the pattern? 080For numbers greater than 4012, reverse the order: :For the last three digits, square the last two:80 x 80 = 6400, carry 6: _ _ _ _ 4 0 0The middle two digits are 8 times the last two (keep thecarry):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 a number between 5000 and 5099. h The first threedigits 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 thenumberchosen (insert zeros when needed): 2 5 0 _ _ X X XExample: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 6So 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 4The middle two digits are 10 times the last two (keep thecarry):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 6099Choose a number between 6000 and 6099. f The first threedigitsare: 3 6 0 _ _ _ _ _ t The next two digits are 12 times the lasttwodigits: h _ _ _ X X _ _ en For the last three digits, square the last two digits in thenumber
  • 165. chosen (insert zeros when u needed): _ _ _ _ _ X X X m bExample: e rto be squared is 6004: The first three digits are: 3 6 0 _ _ _ _ _Thenext 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 6So 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,squarethe last two: 20 x 20 = 400: _ _ _ _ 4 0 0 The middle two digitsare 12times the last two: 12 x 20 = 240 (keep carry): _ _ _ 4 0 _ _ _The firstdigits are 360 + the carry: 360 + 2 = 362: 3 6 2 _ _ _ _ _So 3025 x 3025 = 36,240,400.Squaring numbers between 7000 and 70994 Choose a number between 7000 and 7099. x The first threedigits are: 4 9 0 _ _ _ _ _ 4 The next two digits are 4 times the last twodigits,= with zero added: _ _ _ X X _ _ 1 6 For the last three digits,square thelast two digits in the number chosen (insert zeros when ;needed): _ __ _ _ X X X 1 6 Example: + 4If the number to be squared is 7004: 0 The first three digitsare: 4 9 0_ _ _ _ _ = The next two digits are 4 times the last two, withzeroadded: 5 6: _ _ _ 5 6 _ _ _
  • 166. For the last three digits, square the last two: 4 x 4 = 16: _ _ __016So 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 middletwo digits, add zero to the last two, then add 4 times the lasttwo: 250 + 4 x 25: 250 + 100 = 350 (keep carry): _ _ _ 5 0 _ _ _ Thefirst three digits are 490 + the carry: 490 + 3 = 493: 4 9 3 _ _ ___ So 7025 x 7025 = 49,350,625.Squaring numbers in the hundredsS Choose a number over 100 (keep it low for practice, o thengohigher when expert). 1 1 The last two places will be thesquareof 2 the last two digits (keep any carry) _ _ _ X X. x 1 The firstthreeplaces will be the number plus 1 the last two digits plus anycarry: X X X _ _. 2 = 1 Example: 2 5 If the number to be squared is106: 4Square the last two digits (no carry): 6 x 6 = 36: _ _ _ 3 6 4Add thelast 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 twodigits (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 tosquare the last two digits!
  • 167. Squaring a repeating 6-digit numberChoose a number with repeating 6s .2. The square is made up of: One less 4 than there are digitsin the number; One 3; The same number of 5s as 4s; A final6.Example:If the number to be squared is 666: The square has:one less 4 than digits inthe number 4 4 one 3 3 same number of 5s as 4s 5 5 afinal 6 63. 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 inthe number) 4 4 4 4 one 3 3 same number of 5s as 4s 5 5 55 a final 6 63. So the square of 66666 is 4444355556.Squaring special numbers (3s and final 1)Choose a number with repeating 3s and a final 1. E2. The square is made up of: x One fewer 1 than there arerepeating 3s a 09 m The same number of 5s as there are 1s in the square; pA final 61le:If the number to be squared is 3331: The square has:Two 1s (one fewer thanrepeating 3s) 1 1Next digits: 09 0 9Two 5s (same as 1s in square) 5 5A final 61 6 13. So the square of 3331 is 11,095,561.See the pattern? If the number to be squared is 333331:The square has:Four 1s (one fewer thanrepeating 3s) 1 1 1 1Next digits: 09 0 9
  • 168. Four 5s (same as 1s in square) 5 5 5 5A final 61 6 13. So the square of 333331 is 111,109,555,561.Squaring special numbers (3s and final 2)Choose a number with repeating 3s and a final 2. f2. The square is made up of: t the same number of 1s asthere are repeating 3s; h a zero; e the same number of 2s as there are 1s in the square; n a final 4. um Example: b er to be squared is 3332: The square has:three 1s (number of 3sin number) 1 1 1 a zero 0 three 2s (same as 1s insquare) 2 2 2 a final 4 43. So the square of 3332 is 11,102,224.See the pattern?If the number to be squared is 333332:The square has:five 1s (number of 3sin number) 1 1 1 1 1 a zero 0 five 2s (same as1s in square) 2 2 2 2 2 a final 4 43. So the square of 333332 is 111,110,222,224.These big squares should be quite impressive, and difficultfor others to check unless they have a huge calculator.Squaring special numbers (3s and final 4)S Choose a number with repeating 3s and a final 4. e2. The square is made up of: e the same number of 1s as there are digits in the number; � one fewer 5; ta final 6 heExample: paIf the number to be squared is 3334: tThe square has: tefour 1s (number of digitsr
  • 169. in number) 1 1 1 1 three 5s (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 1s 1 1 1 1 1 1 five 5s 5 5 5 5 5 a final 6 63. So the square of 333334 is 111,111,555,556.Squaring special numbers (3s and final 5)T Choose a number with repeating 3s and a final 5. h2. The square is made up of: e the same number of 1s asthereare repeating 3s in the number; � one more 2 than there aresrepeating 3s; qa final 5. ua Example: re If the number to be squared is 3335: h The square has: as three 1s (same as repeating 3s) 1 1 1 four : 2s (one moretharepeating 3s) 2 2 2 2 a final 5 53. So the square of 3335 is 11,122,225.See the pattern?If the number to be squared is 333335:S five 1s (same as repeating 3s) 1 1 1 1 1 qsix 2s (one morethan repeating 3s) 22 2 2 2 2 a final 5 5ar3. So the square of 333335 is 111,112,222,225.ng special numbers (3s and final 6)Choose a number with repeating 3s and a final 6.2. The square is made up of:the same number of 1s as there are repeating 3s in thenumber;� one 2 � one fewer 8 than there are repeating 3s; a final96.
  • 170. Example:If the number to be squared is 3336: The square has:three 1s (same asrepeating 3s) 1 1 1one 2 2two 8s (one fewer thanrepeating 3s) 8 8 a final 96 9 63. So the square of 3336 is 11,128,896.See the pattern?the number to be squared is 333336: The square has:five 1s (same asrepeating 3s) 1 1 1 1 1one 2 2four 8s (one fewer thanrepeating 3s) 8 8 8 8 a final 96 9 63. So So 333336 x 3333336 = 111,112,888,896.Squaring special numbers (3s and final 7)Choose a number with repeating 3s and a final 7.2. The square is made up of:the same number of 1s as there are repeating 3s in thenumber;� one 3 � one fewer 5 than there are repeating 3s; a final69.Example:If the number to be squared is 3337: The square has:three 1s (same asrepeating 3s) 1 1 1one 3 3two 5s (one fewer thanrepeating 3s) 5 5 a final 69 6 93. So the square of 3337 is 11,135,569.See the pattern?If the number to be squared is 333337: The square has:five 1s (same as
  • 171. repeating 3s) 1 1 1 1 1one 2 2four 5s (one fewer thanrepeating 3s) 5 5 5 5 a final 69 6 93. So So 333337 x 3333337 = 111,113,555,569.Squaring special numbers (3s and final 8)hoose a number with repeating 3s and a final 8.2. The square is made up of:the same number of 1s as there are repeating 3s in thenumber;� one 4 � one fewer 2 than there are repeating 3s; a final44.Example:If the number to be squared is 33338: The square has:four 1s (same asrepeating 3s) 1 1 1 1one 4 4three 2s (one fewer thanrepeating 3s) 2 2 2 a final 44 4 43. 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 1s (same asrepeating 3s) 1 1 1 1 1 1one 4 4five 2s (one fewer thang 3s) 2 2 2 2 2 a final 44 4 43. So So 3333338 x 3333338 = 11,111,142,222,244.Squaring special numbers (3s and final 9)3. So the square of 3333339 = 11,111,148,888,921.Choose a number with repeating 6s and a final 1.2. The square is made up of: � one fewer 4 than there arerepeating 6s
  • 172. 36 two fewer 8s than there are repeating 6s A final 921Example:If the number to be squared is 6661: The square has:two 4s (one fewer than repeating 6s) 4 4 Next digits: 36 3 6one 8 (two fewer than repeating 6s) 8 A final 921 9 2 13. So the square of 6661 is 44,368,921.See the pattern? If the number to be squared is 666661:The square has:four 4s (one fewer thanrepeating 6s) 4 4 4 4 Next digits: 36 3 6three 8s 8 8 8 A final 921 9 2 13. So the square of 666661 is44,4436,888,921.Squaring special numbers (6s and final 2)s Choose a number with repeating 6s and a final 2. a2. The square is made up of: � one fewer 4 than there arerepeating 6s m 38 e number of 2s as 4s in the square a final44Example:If the number to be squared is 6662: The square has:two 4s (one fewer thanrepeating 6s) 4 4Next digits: 38 3 8two 2s (same number as repeating 6s) 2 2A final 44 4 43. So the square of 6662 is 44,382,244.See the pattern? If the number to be squared is 666662:The square has:four 4s (one fewer thanrepeating 6s) 4 4 4 4Next digits: 38 3 8four 2s (same as repeating 6s) 2 2 2 2A final 44 4 43. So the square of 666662 is 444,438,222,244.Squaring special numbers (6s and final 3)
  • 173. I Choose a number with repeating 6s and a final 3. f2. The square is made up of: � one fewer 4 than there arerepeating 6s t 39 h same number of 5s as 4s in the squaree a final 69 n u Example: mber to be squared is 6663: The square has:two 4s (one fewer thanrepeating 6s) 4 4Next digits: 39 3 9two 5s (same number as repeating 6s) 5 5A final 69 6 93. So the square of 6663 is 44,395,569.See the pattern? If the number to be squared is 666663:The square has:four 4s (one fewer thanrepeating 6s) 4 4 4 4Next digits: 39 3 9four 5s (same as repeating 6s) 5 5 5 5A final 69 6 93. So the square of 666663 is 444,439,555,569.Squaring special numbers (6s and final 4)t Choose a number with repeating 6s and a final 4. h2. The square is made up of: r the same number of 4s asrepeating 6s e 0 � one fewer 8 than repeating 6s e a final 96 Example: sIf the number to be squared is 6664: s( The square has: a me number asrepeating 6s) 4 4 4next digit: 38 0two 8s (one fewer than repeating 6s) 8 8a final 96 9 63. So the square of 6664 is 44,408,896.See the pattern?If the number to be squared is 666664:
  • 174. The square has:five 4s (same number asrepeating 6s) 4 4 4 4 4next digit: 0 0four 8s (one fewer thanrepeating 6s) 8 8 8 8 a final 96 9 63. So the square of 666664 is 444,440,888,896.Squaring special numbers (6s and final 5)Choose a number with repeating 6s and a final 5. .2. The square is made up of: same number of 4s asrepeating 6s S same number of 2s as repeating 6s oa final 25tExample: heIf the number to be squared is 6665: The square has:sqthree 4s (same number as repeating 6s) 4 4u4 three 2s (same number as repeatinga6s) 2 2 2 A final 25 2 5re of 6665 is 44,422,225. See the pattern?1. If the number to be squared is 666665:five 4s (same number as repeating 6s) 4 44 4 4 five 2s (same number asrepeating 6s) 2 2 2 2 2 A final 25 252. So the square of 666665 is 444,442,222,225.Squaring special numbers (6s and final 7)Choose a number with repeating 6s and a final 7.2. The square is made up of: The same number of 4sas 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 4s (number of digitsin number) 4 4 4 4 three 8s (one fewer) 8 8 8 a final 93. So the square of 6667 is 44448889. See the pattern? If thenumber to besquared is 667:The square has:ring special numbers (6s and final 8)Choose a number with repeating 6s and a final 8.2. The square is made up of:the same number of 4s as there are repeating 6s in thenumber; � one 6 thesame number of 2s as repeating 6s; a final 4.Example:If the number to be squared is 6668: The square has:three 4s (same asrepeating 6s) 4 4 4one 6 6three 2s (same number asrepeating 3s) 2 2 2a final 4 43. So the square of 6668 is 44,462,224.See the pattern?3 If the number to be squared is 666668: . The square has: Sofive 4s (same number asSrepeating 6s) 4 4 4 4 4 o 6 one 6 6 66five 2s (same number as6repeating 6s) 2 2 2 2 2 6 8a final 4 4x 6 66668 = 444,446,222,224.Squaring special numbers (6s and final 9)
  • 176. Choose a number with repeating 6s and a final 9.2. The square is made up of:the same number of 4s as there are repeating 6s;a 7 � one fewer 5 than there are repeating 6s; A final 61.Example:If the number to be squared is 6669: The square has:same number of 4s as repeating 6s: 4 4 4a 7 7 one fewer 5than repeating 6s 5 5a final 61 6 13. 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 4s as 6s: 4 4 4 4 4a 7 7 one fewer 5 thanrepeating 6s 5 5 5 5a final 61 6 13. So 666,669 x 666,669 = 44,447,555,561. Use the pattern toamaze yourfriends with your multiplying abilities.Squaring special numbers (9s and final 1)t h Choose a number with repeating 9s and a final 1. e2. The square is made up of: � one fewer 9 than there arerepeating 9s same number of 0s as there are 9s in the square A final 81Example:If the number to be squared is 9991: The square has:One fewer 9 than therepeating 9s: 9 9 82 8 2 same number of 0s as9sin the square 0 0 a final 81 8 13. So 9991 x 9991 = 99820081.See the pattern?If the number to be squared is 999991: The square has:one fewer 9 than therepeating 9s: 9 9 9 9 82 8 2 same number of 0s as 9sin the square 0 0 0 0 a final 81 8 13. So 999991 x 999991 = 999982000081.
  • 177. Those big products ought to impress your friends, and theywill needa BIG calculator to keep up with you!Squaring special numbers (9s and final 2)Choose a number with repeating 9s and a final 2.2. The square is made up of: � one fewer 9 than there arerepeating 9sthe same number of 0s as there are 9s in the square A final64Example:If the number to be squared is 9992: The squareone fewer 9 than the9 9 84 8 4 same number of 0s as 9sin the square 0 0 a final 64 6 43. So 9992 x 9992 = 99840064. See the pattern?If the number to be squared is 999992: The square has:one fewer 9 than therepeating 9s: 9 9 9 9 84 8 4 same number of 0s as 9sin the square 0 0 0 0 a final 64 6 43. So 999992 x 999992 = 999984000064. With a little practiceyou willbe finding these huge products with ease.Squaring special numbers (9s and final 3)SeetChoose a number with repeating 9s and a final 3. h2. The square is made up of: � one fewer 9 than there arerepeating9s e 86 pthe same number of 0s as there are 9s in the squarea A final 49 ttExample: er If the number to be squared is 9993: nThe square has: ?
  • 178. one fewer 9 than therepeating 9s: 9 9 86 8 6 same number of 0s as I 9s f in thesquare 0 0 a final 49 4 9 t h3. So 9993 x 9993 = 99860049. e number to be squared is 999993: The square has:one fewer 9 than therepeating 9s: 9 9 9 9 86 8 6 same number of 0s as 9sin the square 0 0 0 0 a final 49 4 93. So 999993 x 999993 = 999986000049. Using this patternyou willbe able to square these large numbers with ease.Squaring special numbers (9s and final 4)Choose a number with repeating 9s and a final 4.2. The square is made up of: � one fewer 9 than there arerepeating 9s 88 the same number of 0s as there are 9s inthe square A final 36Example:If the number to be squared is 9994: The square has:one fewer 9 than therepeating 9s: 9 9 88 8 8 same number of 0s as9sin the square 0 0 a final 36 3 63. So 9994 x 9994 = 99880036.See the pattern?If the number to be squared is 999994: The square has: one fewer 9 than therepeating 9s: 9 9 9 9 88 8 8 same number of 0s as 9sin the square 0 0 0 0 a final 36 3 63. So 999994 x 999994 = 999988000036.Squaring special numbers (9s and final 5)Choose a number with repeating 9s and a final 5.2. The square is made up of: same number of 9s as thereare repeating 9s same number of 0s a final 25Example:
  • 179. If the number to be squared is 9995: The square has:same number of 9s as repeating 9s: 9 9 9same number of 0s as 9s in the square 00 0 a final 25 2 53. So 9995 x 9995 = 99900025. See the pattern?1. If the number to be squared is 999995:same number of 9s as repeating 9s: 9 9 9 9 samenumber of 0s as 9s in the square 0 0 0 0 afinal 25 2 53. So 999995 x 999995 = 999990000025.Squaring special numbers (9s and final 6)h o o se a number with repeating 9s and a final 6.2. The square is made up of: same number of 9s as thereare repeating 9s a 2 � one fewer 0 than repeating 9s a final16Example:If the number to be squared is 9996: The square has:same number of 9sas repeating 9s: 9 9 9a22one fewer 0 than thereare 9s in the square 0 0 a final 16 1 63. So 9996 x 9996 = 99920016.See the pattern?If the number to be squared is 999996: The square has:same number of 9sas repeating 9s: 9 9 9 9 9a22one fewer 0 than thereare 9s in the square 0 0 0 0 a final 16 1 63. So 999996 x 999996 = 999992000016.Squaring special numbers (9s and final 7)Afi n Choose a number with repeating 9s and a final 7. a
  • 180. 2. The square is made up of: l the same number of 9s asthere are repeating 9s 9 4 the same number of 0s as thereare 9s in thesquare Example:If the number to be squared is 9997: The square has:same number of 9s as there are repeating9s: 9 9 9 4 4 same number of 0s as 9sin the square 0 0 0 a final 9 93. So 9997 x 9997 = 99940009. See the pattern?If the number to be squared is 999997: The square has:same number of 9s as there are repeating 9s:9 9 9 9 9 4 4 same number of 0s as 9s inthe square 0 0 0 0 0 a final 9 93. So 999997 x 999997 = 999994000009. Learn the pattern andits easy!Squaring special numbers (9s and final 8)Choose a number with repeating 9s and a final 8.2. The square is made up of:the same number of 9s as there are repeating 9sthe same number of 0s as there are 9s in the square A final4Example:If the number to be squared is 9998: The square has:same number of 9s as there are repeating9s: 9 9 9 6 6 same number of 0s as 9s0 a final 4 43. So 9998 x 9998 = 99,960,004.See the pattern?If the number to be squared is 999998: The square has:same number of 9s as there are repeating 9s:9 9 9 9 9 6 6 same number of 0s as 9s inthe square 0 0 0 0 0 a final 4 43. So 999997 x 999997 = 999,996,000,004.
  • 181. Subtract from a repeating 8s number, divide by 9, subtract102Select a number between 2 and 9 digits made up of 8s. 1Subtract a number equal to 9 minus the number of digitsin the number you have selected. Divide by 9. The answerwill have one less digit than the original number in theseries 98765432. Subtract 10.Example:If the repeating number selected has 4 digits: 8888: Subtract5(9 minus the number of digits) and divide by 9. The answerwillbe 987 (1 less digit in sequence than the 4 digits in theselectednumber). 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 the6 digits in the selected number). Subtract 10: 98765 - 10 =98755. So (888888-3) / 9 - 10 = 98755.Extend this exercise by changing the last number tosubtract,producing a multitude of easily answeredproblems. Subtract from a repeating 1s number, divide by 9,subtractSelect a number between 2 and 10 digits made up of 1s.Subtract a number equal to the number of digits in thenumber you have selected. Divide by 9. The answer willhave one less digit than the original number in the series123456789. Subtract 21.Example:
  • 182. If the repeating number 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 9The 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 is12345679011 13 digit number, answer is d 12345679012214 digit number, answer is 1234567901233 t 15 digit number, answer is 12345679012344 h e and ? exercises using this basic pattern by changing step 4 to addorsubtract other numbers.Subtracting the squares of two numbersSelect a 2-digit number and then choose a number two largeror two smaller. Multiply the middle number by 4.Example:If the the first number selected is 31: Choose 29. Multiply thenumber 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 thenumber 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 thenumber between them by 4: 4 x 74 = 2964. So (75x75) - (73x73) = 296.Adding even numbers from two through a selected 2-digiteven number
  • 183. Divide the even number by 2 (or multiply by 1/2). . Multiplythisresult by the next number. T Example: he 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). ext 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 leftto 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 numbersfrom 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 22by 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 is462.Squaring numbers made up of threesChoose a a number made up of threes.2. The square is made up of:a. one fewer 1 than there are repeating 3sb. zeroc. one fewer 8 than there are repeating 3s (same as the 1s inthe square)d. nine.
  • 184. Example:If the number to be squared is 3333: The square of thenumber has:three 1 s (onefewer than digits in number) 1 1 1 _ _ _ _ _ next digit is 0_ _ _ 0 _ _ _ _ three 8s (same number as 1s) _ _ _ _ 8 8 8_ a final 9 _ _ _ _ _ _ _ 93. So 3333 x 3333 = 11108889.See the pattern?If the number to be squared is 333:The square of the number has:two 1snext digit is 0two 8sa final 9. So 333 x 333 = 110889.11________0_______88_______9Multiplying two 2-digit numbers(same 1st digit) S oSelect two 2-digit numbers with the same first digit. 42Multiply their second digits (keep the carry). _ _ _ X x Multiplythesum of the second digits by the first digit, 4 add the carry(keepthe carry). _ _ X _ 5= Multiply the first digits (add the carry). X X _ _ 18 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) _ _ _ 0Multiply 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 = 1818__If See the pattern? the 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 bythe first (add carry): 2 + 7 = 9; 6 x 9 = 54; 54 + 1 = 55 (keepcarry) _ _ 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 firstdigits and add one second: X X _ _.Multiply the seconddigits 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 1stdigits, 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 1stdigits, add one 2nd. 9x1 = 9, 9+3 = 12. 1 2 _ _Multiply the 2nd digits. 3x3 = 9 _ _ 0 94. So 93 x 13 = 1209.Multiplying two 2-digit numbersM (same 1st digit, 2nd digits sum to 10) u
  • 186. lt Both numbers should have the same first digit. Choosesecond digits whose sum is 10. ply the first digit by one number greater than itself; thisnumberwill be the first part of theanswer: X X _ _.Multiply the two second digits together; the productwill 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 thefirst 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 (multiplythe first digit by one number greater than itself): the first partof the answer is 2 0 _ _. 7 x 3 = 21 (multiply the two seconddigits together); the last part of the answer is _ _ 2 1. So 47 x43 = 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 (multiplythe first digit by one greater), the first part of the answer is 42_ _. 2 x 8 = 16 (multiply the two second digits together); thelast part of the answer is _ _ 1 6. So 62 x 68 = 4216.Multiplying two selected 3-digit numbersT (middle digit 0) he Select a 3-digit number with a middle digit of 0. l Choose amultiplier with the same first two digits, whose third digitsums 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 of the numbers: o _ X _ _ _ or _ _ X _ _ _. di The next digit is zero: _ _ 0 _ _ or _ _ _ 0 _ _. g
  • 187. its will be the product of the third digits: _ _ _ X X or _ _ __ X X.Example:If the first number is 407, choose 403 as the secondnumber (same first digits, second digits add to 10). 4 x 4= 16 (square the first digit): 1 6 _ _ _ _. The next digit willbe the first digit of the numbers: _ _ 4 _ _ .The next digit is zero: _ _ 0 _ _ . 7 x 3 = 21 (the last two digitswill 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 (squarethe first digit): 4 _ _ _ _. The next digit will be the first digit ofthe numbers: _ 2 _ _ _ .The next digit is zero: _ _ 0 _ _ . 1 x 9 = 09 (the last two digitswill 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 amultiplier with the same first two digits, whose third digitsums 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 thenumber (keep carry):_ X _ _ _ or _ _ X _ _ _.The first digits will be thesquare of the first digit plus thecarry: X _ _ _ _ or X X _ _ _ _.As you determine the digits in the answer fromright to left, repeat them to yourself at each stepuntil you have the whole answer.Example:
  • 188. If the first number is 814, choose 816 as the secondnumber (same first digits, second digits add to 10).4 x 6 = 24 (multiply the first digits) - last two digits: _ _ _ _ 24.The third digit from the right is 2: _ _ 2 _ _ . D K TIWARI CFA ~MODERATOR~ SBI+ALL BANK PO ASPIRANTS COMMUNITY

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