4. Positive Integers
Non – Negative Integers
- Natural Numbers
- Whole Numbers
Prime Numbers
- Divisible by 1 and by the number itself
- Only 2 factors
2, 3, 5, 7, 11, 13, 17, 19 . . .
Composite Numbers
- Except Prime Numbers
1 – Neither Prime nor Composite
2 – Only Even Prime number
7. Divisibility Tests
Divisible by 2
- All even numbers
- Numbers end with 0, 2, 4, 6, 8
Divisible by 3
- Sum of the digits divisible by 3
Example –
176
1 + 7 + 6 = 14 not divisible
24871
2 + 4 + 8 + 7 + 1 = 22 not divisible
14913
1 + 4 + 9 + 1 + 3 = 18 divisible
8. Divisibility Tests
Divisible by 4
- last 2 digits divisible by 4
Example –
226 - 26 / 4 not divisible
1248 - 48 / 4 divisible
Divisible by 5
- Last digit 0 or 5
Divisible by 6
- Divisible by both 2 and 3
9. Divisibility Tests
Divisible by 8
- last 3 digits divisible by 8
Example –
12226
226 / 8 not divisible
21248
248 / 8 divisible
10. Divisible by 9
- Sum of the digits divisible by 9
Example –
176
1 + 7 + 6 = 14 not divisible
24871
2 + 4 + 8 + 7 + 1 = 22 not divisible
14913
1 + 4 + 9 + 1 + 3 = 18 divisible
Divisible by 10
- Last digit 0
Divisibility Tests
11. Divisibility Tests
Divisible by 11
- The difference between Sum of odd terms
and even terms = 0 or divisible by 11
Example
156348
1 + 6 + 4 = 11
5 + 3 + 8 = 16
Difference = 16 – 11 = 5
not divisible
12. Divisibility Tests
Divisible by 7
7x7 = 49
49 + 1 = 50
x5 and Add
Example 1
1321
132 1x5
5
137
13 7x5
35
48
Not Divisible
Example 2
16548
1654 8x5
40 1694
169 4x5
20 189
18 9x5
45 63
Divisible
14. Divisible by 12
- Divisible by both 3 and 4
To select 2 numbers to check divisibility
- Product should be the original number
- The numbers should be co – prime
Divisible by 18
- Divisible by 2 and 9
Divisible by 24
- Divisible by 3 and 8
Divisibility Tests
15. Divisible by 35
- Divisible by 5 and 7
Divisible by 36
- Divisible by 4 and 9
Divisibility Tests
16. 1. What is the minimum number to be added to make
329611 as divisible by 6
a) 1 b) 3 c) 5 d) 7
divisible by 6
divisible by 2 and 3
/3
3 + 2 + 9 + 6 + 1 + 1
= 22
From options
a 22 + 1 = 23 not divisible
b 22 + 3 = 25 not divisible
c 22 + 5 = 27 divisible
/2
329611 + 5 = 329616
even number
so divisible by 2
Answer : c
17. 2. 1358k24679 is divisible by 11. Find the value of k
a) 0 b) 1 c) 4 d) 9
divisible by 11
1358k24679
1 + 5 + k + 4 + 7 =
3 + 8 + 2 + 6 + 9 =
28 – (17 + k) = 0 or divisible by 11
so k = 0
answer : a
17 + k
28
18. 3. 4231k4 is divisible by 24. Find the value of k
a) 7 b) 4 c) 6 d) 1
divisible by 24
- divisible by 3 and 8
4 + 2 + 3 + 1 + k + 4 = 14 + k
from options
a 14 + 7 = 21 divisible by 3
174 / 8 not divisible by 8
b 14 + 4 = 18 divisible by 3
144 / 8 divisible by 8
answer : b
19. 4.11564k4 is divisible by 36. Find the value of k
a) 6 b) 2 c) 9 d) 7
divisible by 36
- divisible by 4 and 9
1 + 1 + 5 + 6 + 4 + k + 4 = 21 + k
from options
a 21 + 6 = 27 divisible by 9
64 / 4 = 16 divisible by 4
answer : a
20. 5. 996ab is divisible by 80. What is (a + b) ?
a) 3 b) 5 c) 6 d) 8
divisible by 80
996ab
b must be 0
996a is divisible by 8
Option a and b 3 & 5 odd numbers
not divisible by 8
option c 966 / 8
not divisible by 8
so a = 8 968 / 8 divisible by 8
a = 8 ; b =0 a + b = 8
answer : d
21. 6. 477 is not divisible by
a) 53 b) 3 c) 6 d) 9
477 is odd number
not divisible by 6
477 / 53 = 9 divisible by 53
4 + 7 + 7 = 18 divisible by 3 and 9
answer : c
22. 7. The product of 4 consecutive even numbers is
always divisible by:
a) 600 b) 768 c) 864 d) 384
Product of first 4 consecutive even numbers
= 2 * 4 * 6 *8
= 384
answer : d
23. 8. What is the largest 4 digit number exactly divisible
by 88?
a) 9944 b) 9999 c) 9988 d) 9900
Divisible by 88
- Divisible by 8 and 11
all 4 options divisible by 11
a 944 / 8 = 118 divisible by 8
b odd number not divisible by 8
c 988 / 8 not divisible by 8
answer : a
24. (a^n) – (b^n)
is divided by (a-b) for ∀n∈ N
is divided by(a+b) for all even integers of N
(a^n) + (b^n)
is divisible by (a+b) for all odd integers of N
9. Which of the following numbers will completely
divide ((49^15) - 1) ?
a) 11 b) 7 c) 8 d) 9
((49^15) - 1) = (7^30) – 1
It is divisible by 7+1
8
25. 10. It is being given that (232 + 1) is completely divisible
by a whole number. Which of the following numbers is
completely divisible by the number?
a) (216 + 1) b) (216 - 1) c) (7 x 223) d) (296 + 1)
(296 + 1) = (232 ) 3+ 1
So Its divisible by (296 + 1)
26. Unit Digit
Find the unit digit of the sum
263 + 541 + 275 + 562 + 758
Add only the unit digits
= 19
Unit digit = 9
Find the unit digit of the product
543 * 548 * 476 * 257
Multiply only unit digits
unit digit = 8
27. Unit Digit
what is the unit digit of
1237 - 4563 + 6214 + 8541 - 5612
a) 5 b) 3 c) 8 d) 7
Answer : d
Find the unit digit of the product
567 x 142 x 263 x 429 x 256
a) 5 b) 3 c) 8 d) None of these
Answer : c
Find the unit digit of product of the prime numbers up to 50.
a) 0 b) 1 c) 2 d) 5
2 * 3 * 5 * …
= 0
Answer : a
28. Unit digit of 2^ 156 = ?
2^1 = 2
2² = 4
2³ = 8
2⁴ = 16
Unit Digit
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
Unit digit of 2^ any number is one among
2, 4, 8 and 6
Divide the power by 4
If divisible unit digit 2⁴ = 6
If remainder 1 unit digit 2^1 = 2
If remainder 2 unit digit 2² = 4
If remainder 3 unit digit 2³ = 8
Unit digit of 2^ 156
56 / 4
divisible
so unit digit is 6
29. 3^1 = 3
3² = 9
3³ = 27
3⁴ = 81
Unit Digit
Unit digit of 3^ any number is one among
3, 9, 7 and 1
Divide the power by 4
If divisible unit digit 3⁴ = 1
If remainder 1 unit digit 3^1 = 3
If remainder 2 unit digit 3² = 9
If remainder 3 unit digit 3³ = 7
30. 4^1 = 4
4² = 16
4³ = 64
4⁴ = 256
Unit Digit
Unit digit of 4^ any number is 4 or 6
If the power is odd number
Unit digit is 4
If the power is even number
Unit digit is 6
5^1 = 5
5² = 25
5³ = 125
Unit digit is always 5
6^1 = 6
6² = 36
6³ = 216
Unit digit is always 6
31. 7^1 = 7
7² = 49
7³ = __3
7⁴ = __1
Unit Digit
Unit digit of 7^ any number is one among
7, 9, 3 and 1
Divide the power by 4
If divisible unit digit 7⁴ = 1
If remainder 1 unit digit 7^1 = 7
If remainder 2 unit digit 7² = 9
If remainder 3 unit digit 7³ = 3
32. 8^1 = 8
8² = 64
8³ = __2
8⁴ = __6
Unit Digit
Unit digit of 8^ any number is one among
8, 4, 2 and 6
Divide the power by 4
If divisible unit digit 8⁴ = 6
If remainder 1 unit digit 8^1 = 8
If remainder 2 unit digit 8² = 4
If remainder 3 unit digit 8³ = 2
33. 9^1 = 9
9² = 81
9³ = __9
9⁴ = __1
Unit Digit
Unit digit of 9^ any number is 9 or 1
If the power is odd number
Unit digit is 9
If the power is even number
Unit digit is 1
1^1 = 1
1² = 1
1³ = 1
Unit digit is always 1
0^1 = 0
0² = 0
0³ = 0
Unit digit is always 0
34. Unit Digit
2, 3, 7 and 8
Divide the power by 4
4 and 9
Power odd / even number
0, 1, 5 and 6
The number itself
35. 1. What is the digit in the unit place of the number
represented by
(79^5 – 35^8)?
a) 4 b) 3 c) 2 d) 1
(79^5 – 35^8)
9^5 - 5^8
=
= 4
9^5
Power is odd number
So Unit digit is 9
5^8
Unit digit is 5
9 5
-
Answer : a
36. 2. Find the unit digit of
352^4 + 43^24 + 4137^ 754 + 689 ^ 561
a) 5 b) 6 c) 7 d) 8
352^4 + 43^24 + 4137^ 754 + 689 ^ 561
= 2^4 + 3^24 + 7^ 754 + 9 ^ 561
=
2^4 = 16 unit digit 6
3^24 Divide the power by 4
24 / 4 divisible
So 3^4 Unit digit 1
7^ 754 Divide the power by 4
54 / 4 = 13 remainder 2
So 7² = 49 Unit digit 9
9 ^ 561
Power 561 is odd number
Unit digit 9
6
Answer : a
1 9 9 =
+
+
+ 25
Unit digit 5
37. 3. Find the unit digit of the product
(2012^2012)*(2013^2013)*(2014^2014)*(2015^2015)
a) 0 b) 2 c) 3 d) 5
(2012^2012) * (2013^2013) * (2014^2014) * (2015^2015)
(2^2012) * (3^2013) * (4^2014) * (5^2015)
(2^2012)
Divide the power by 4
= 12 / 4
Divisible
Unit digit is 6
(3^2013)
Divide the power by 4
= 13 / 4
Remainder 1
Unit digit is 3
(4^2014)
Power is even number
Unit digit is 6
(5^2015)
Unit digit is 5
6 6
3 5
*
*
* 0
Answer : a
38. 4. Find the unit digit of 4! + 10!
a) 0 b) 4 c) 5 d) 6
1! = 1
2! = 1 x 2 = 2
3! = 1 x 2 x 3 = 6
4! = 1 x 2 x 3 x 4 = 24
5! = 1 x 2 x 3 x 4 x 5 = 120
Next all numbers factorials Unit digit is 0
4! + 10!
4 + 0 = 4
Answer : b
39. 5. Find the unit digit of 0! + 3! + 200!
a) 0 b) 5 c) 6 d) 7
0! = 1
3! = 1 x 2 x 3 = 6
0! + 3! + 200!
1 + 6 + 0 = 7
Answer : d
40. Number of zeroes at the end
2 x 5 = 10
2 x 2 x 5 x 5 = 100
2 x 2 x 2 x 5 x 5 = 200
2 x 2 x 5 x 5 x 5 = 500
2 x 2 x 2 x 5 x 5 x 5 = 1000
So number of 2 and 5 pairs = Number of zeroes at the end
41. 1. Number of zeros at the end of
1200 + 1265000 + 1500000 + 9213000 + 2100 is
a) 2 b) 3 c) 4 d) 5
1200
1265000
1500000
9213000
2100
300
Answer : a
42. 2. Calculate the number of zeros at the end of 63!
a) 12 b) 15 c) 21 d) 14
63 ! = 1 x 2 x 3 x 4 x 5 x . . . x 63
63
5
12
5
Answer : d
= 12
= 2
14
43. 3. Calculate number of zeros at the end of 74! & 75!
a) 16 & 17 b) 16 & 18 c) 17 & 18 d) 18 & 19
74
5
14
5
= 14
2
=
16
75
5
15
5
Answer : b
= 15
3
=
18
44. 4. Calculate the number of zeros at the end of 625!
a) 155 b) 150 c) 125 d) 156
625
5
125
5
25
5
5
5
= 125
25
=
=
5
1
156
=
Answer : d
45. 5. Number of zeros at the end of (100!) X (120!)
a) 28 b) 52 c) 672 d) 252
100
5
20
5
= 20
4
=
24
120
5
24
5
= 24
4
=
28
24 + 28
= 52
Answer : b
46. 6. The number of zeros at the end of the product of all prime numbers
between 1 and 1111 is?
a) 0 b) 1 c) 225 d) 275
Only one 2 and
Only one 5
So only one zero
Answer : b
47. 7. The number of zeros at the end of the product of
74 x 75 x 76x 77 x 78 x 79 x 80 is?
a) 1 b) 2 c) 3 d) 4
5s in 75 and 80
75 = 3 x 25
= 3 x 5 x 5
80 = 16 x 5
Only 3 number of 5s
and more number of 2s
So 3 zeroes
Answer : c
48. 8. The number of zeros at the end of the product of
2 x 5 x 8 x 25 x 75 is?
a) 1 b) 2 c) 3 d) 4
5s in 5, 25 and 75
5
25 = 5 x 5
75 = 3 x 5 x 5
5 number of 5s
2
8 = 2 x 2 x 2
Only 4 number of 2s
So 4 zeros
Answer : d
49. 1. What is the highest power of 3 that can divide 100!?
100
3
33
3
11
3
3
3
= 33
1
=
48
= 11
=
3
Maximum Power
50. 2. What is the maximum power of 7 contained in 120!?
120
7
17
7
= 17
19
= 2
51. 3. What is the highest power of 4 that can divide 125!?
125
2
62
2
31
2
15
2
= 62
7
=
= 31
=
15
125! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x . . . X 125
2 x 3
4 = 2²
7
2
3
2
2^119
2² ^ ?
= 119 / 2
= 59
3
=
1
=
119
52. 4. What is the highest power of 12 that can divide 116!?
12 = 3 x 4
4 = 2²
12 = 2² x 3
116
2
58
2
29
2
14
2
= 58
7
=
= 29
=
14
7
2
3
2
2^112
2² ^ ?
= 112 / 2
= 56
3
=
1
=
112
54. 5. What is the highest power of 8 that can divide 125!?
8 = 2 x 4
= 2 x 2 x 2
= 2³
125
2
62
2
31
2
15
2
= 62
7
=
= 31
=
15
7
2
3
2
2^119
2³ ^ ?
= 119 / 3
= 39
3
=
1
=
119
55. 6. What is the maximum power of 18 in 210!?
a) 102 b) 51 c) 206 d) 103
18 = 2 x 9
= 2 x 3 x 3
= 2 x 3²
210
2
105
2
52
2
26
2
= 105
13
=
= 52
=
26
13
2
6
2
3
2
2^206
6
=
3
=
206
= 1
