100 Number System MCQs with Answers & Analytical Explanations is a comprehensive practice set designed for students and competitive-exam aspirants. These questions cover important concepts such as natural numbers, whole numbers, integers, rational and irrational numbers, prime numbers, composite numbers, divisibility rules, HCF, LCM, remainders, factors, multiples, and numerical properties.
Each question includes the correct answer along with an analytical explanation, helping learners understand the method rather than simply memorize the answer. This practice set can be useful for SSC, Banking, Railway, UPSC, State PSC, Defence, entrance examinations, and other competitive exams.
100 Number System MCQs with Answers & Analytical Explanations
1. What is the smallest natural number?
A) 0
B) 1
C) 2
D) -1
Answer: B) 1
Explanation: Natural numbers generally begin with 1, 2, 3, 4, …. Therefore, 1 is the smallest natural number.
2. Which of the following is a whole number?
A) -3
B) -1
C) 0
D) -5
Answer: C) 0
Explanation: Whole numbers are 0, 1, 2, 3, …. They do not include negative numbers or fractions.
3. Which of the following is an integer?
A) 2.5
B) 3/4
C) -7
D) 1.25
Answer: C) -7
Explanation: Integers include negative numbers, zero, and positive whole numbers: …, -2, -1, 0, 1, 2, …
4. Which number is irrational?
A) 3/5
B) 0.25
C) √2
D) 7
Answer: C) √2
Explanation: √2 cannot be expressed as a ratio of two integers and its decimal expansion is non-terminating and non-repeating. Hence, it is irrational.
5. Which of the following is a prime number?
A) 21
B) 27
C) 29
D) 39
Answer: C) 29
Explanation: 29 has only two positive factors: 1 and 29. Therefore, it is prime.
6. What is the only even prime number?
A) 0
B) 1
C) 2
D) 4
Answer: C) 2
Explanation: 2 is divisible only by 1 and itself. Every other even number has at least three factors.
7. Which of the following is a composite number?
A) 13
B) 17
C) 19
D) 21
Answer: D) 21
Explanation: 21 = 3 × 7, so it has factors other than 1 and itself. Hence, it is composite.
8. What is the smallest composite number?
A) 1
B) 2
C) 3
D) 4
Answer: D) 4
Explanation: 1 is neither prime nor composite, while 2 and 3 are prime. 4 = 2 × 2, making it the smallest composite number.
9. What is the HCF of 24 and 36?
A) 6
B) 8
C) 12
D) 18
Answer: C) 12
Explanation: Factors common to 24 and 36 include 1, 2, 3, 4, 6 and 12. The greatest is 12.
10. What is the LCM of 12 and 18?
A) 24
B) 30
C) 36
D) 48
Answer: C) 36
Explanation:
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 36.
11. What is the HCF of 48 and 72?
A) 12
B) 18
C) 24
D) 36
Answer: C) 24
Explanation:
48 = 2⁴ × 3
72 = 2³ × 3²
Common factors give 2³ × 3 = 24.
12. What is the LCM of 15 and 20?
A) 30
B) 40
C) 50
D) 60
Answer: D) 60
Explanation:
15 = 3 × 5
20 = 2² × 5
LCM = 2² × 3 × 5 = 60.
13. If HCF of two numbers is 6 and their LCM is 180, and one number is 30, what is the other number?
A) 24
B) 36
C) 42
D) 48
Answer: B) 36
Explanation: For two positive integers:
First number × Second number = HCF × LCM
30 × x = 6 × 180
x = 1080 ÷ 30 = 36.
14. What is the remainder when 125 is divided by 7?
A) 4
B) 5
C) 6
D) 3
Answer: B) 6
Explanation: 7 × 17 = 119. Therefore, 125 − 119 = 6.
15. What is the remainder when 1000 is divided by 9?
A) 0
B) 1
C) 2
D) 3
Answer: B) 1
Explanation: 1000 = 9 × 111 + 1. Therefore, the remainder is 1.
16. Which number is divisible by 3?
A) 124
B) 235
C) 342
D) 451
Answer: C) 342
Explanation: Sum of digits = 3 + 4 + 2 = 9, which is divisible by 3. Hence, 342 is divisible by 3.
17. Which number is divisible by 9?
A) 234
B) 315
C) 427
D) 512
Answer: B) 315
Explanation: 3 + 1 + 5 = 9, which is divisible by 9.
18. Which number is divisible by 11?
A) 121
B) 123
C) 145
D) 156
Answer: A) 121
Explanation: For 121, the alternating digit difference is:
(1 + 1) − 2 = 0.
Therefore, 121 is divisible by 11.
19. What is the smallest number divisible by 2, 3 and 5?
A) 15
B) 20
C) 30
D) 60
Answer: C) 30
Explanation: LCM(2, 3, 5) = 30.
20. What is the smallest number divisible by 4, 6 and 8?
A) 12
B) 18
C) 24
D) 48
Answer: C) 24
Explanation:
4 = 2²
6 = 2 × 3
8 = 2³
LCM = 2³ × 3 = 24.
21. What is the unit digit of 7²?
A) 7
B) 9
C) 1
D) 3
Answer: B) 9
Explanation: 7² = 49. Therefore, the unit digit is 9.
22. What is the unit digit of 7³?
A) 1
B) 3
C) 7
D) 9
Answer: B) 3
Explanation: 7³ = 343, so the unit digit is 3.
23. What is the unit digit of 2⁵?
A) 2
B) 4
C) 6
D) 8
Answer: B) 2
Explanation: 2⁵ = 32. Therefore, the unit digit is 2.
24. What is the unit digit of 3⁴?
A) 1
B) 3
C) 7
D) 9
Answer: A) 1
Explanation: 3⁴ = 81. Hence, the unit digit is 1.
25. What is the unit digit of 9⁵?
A) 1
B) 5
C) 7
D) 9
Answer: D) 9
Explanation: Powers of 9 alternate between 9 and 1. Odd powers have unit digit 9.
26. What is the remainder when 2¹⁰ is divided by 3?
A) 0
B) 1
C) 2
D) 3
Answer: B) 1
Explanation: 2 ≡ -1 (mod 3). Therefore:
2¹⁰ ≡ (-1)¹⁰ = 1.
27. What is the remainder when 5⁴ is divided by 4?
A) 0
B) 1
C) 2
D) 3
Answer: B) 1
Explanation: 5 ≡ 1 (mod 4). Thus, 5⁴ ≡ 1⁴ = 1.
28. What is 0 × 999?
A) 999
B) 1
C) 0
D) Undefined
Answer: C) 0
Explanation: Any number multiplied by zero is zero.
29. Which of the following is neither prime nor composite?
A) 0
B) 1
C) 2
D) 3
Answer: B) 1
Explanation: A prime number has exactly two positive factors, while a composite number has more than two. 1 has only one factor, so it is neither.
30. What is the sum of the first 10 natural numbers?
A) 45
B) 50
C) 55
D) 60
Answer: C) 55
Explanation:
Sum = n(n + 1)/2
= 10 × 11 / 2
= 55.
31. What is the sum of the first 20 natural numbers?
A) 200
B) 210
C) 220
D) 230
Answer: B) 210
Explanation:
20 × 21 ÷ 2 = 210.
32. What is the sum of the first 10 odd numbers?
A) 90
B) 100
C) 110
D) 120
Answer: B) 100
Explanation: The sum of the first n odd numbers is n². Therefore, 10² = 100.
33. What is the sum of the first 10 even numbers?
A) 100
B) 110
C) 120
D) 90
Answer: B) 110
Explanation: First 10 even numbers are 2 through 20.
Sum = 10(10 + 1) = 110.
34. How many prime numbers are there between 1 and 20?
A) 6
B) 7
C) 8
D) 9
Answer: C) 8
Explanation: The primes are 2, 3, 5, 7, 11, 13, 17, 19.
35. What is the average of the first five natural numbers?
A) 2
B) 3
C) 4
D) 5
Answer: B) 3
Explanation:
Average = (1 + 2 + 3 + 4 + 5) ÷ 5
= 15 ÷ 5 = 3.
36. What is the greatest two-digit prime number?
A) 89
B) 91
C) 97
D) 99
Answer: C) 97
Explanation: 97 has no factors other than 1 and 97. Therefore, it is the greatest two-digit prime.
37. What is the smallest three-digit prime number?
A) 101
B) 103
C) 107
D) 109
Answer: A) 101
Explanation: 101 is divisible only by 1 and itself, making it the smallest three-digit prime.
38. What is the greatest three-digit number divisible by 8?
A) 992
B) 996
C) 998
D) 999
Answer: A) 992
Explanation: 999 ÷ 8 leaves remainder 7. Therefore, subtract 7:
999 − 7 = 992.
39. What is the smallest four-digit number divisible by 15?
A) 1000
B) 1005
C) 1010
D) 1015
Answer: B) 1005
Explanation: 1000 is not divisible by 15. The next multiple is 1005 = 15 × 67.
40. What is the greatest number that divides 43, 91 and 183 leaving the same remainder?
A) 4
B) 6
C) 8
D) 12
Answer: A) 4
Explanation: If the same remainder is left, the divisor must divide the differences:
91 − 43 = 48
183 − 91 = 92
HCF(48, 92) = 4.
41. What is the HCF of 17 and 19?
A) 1
B) 2
C) 17
D) 19
Answer: A) 1
Explanation: 17 and 19 are distinct prime numbers, so their only common factor is 1.
42. What is the LCM of two distinct prime numbers 7 and 11?
A) 18
B) 77
C) 49
D) 121
Answer: B) 77
Explanation: For distinct primes, LCM is their product:
7 × 11 = 77.
43. If two numbers are co-prime, their HCF is:
A) 0
B) 1
C) 2
D) Their product
Answer: B) 1
Explanation: Co-prime numbers have no common factor other than 1.
44. Which pair is co-prime?
A) 8, 12
B) 14, 21
C) 15, 28
D) 18, 24
Answer: C) 15, 28
Explanation: Factors of 15 are 1, 3, 5, 15; factors of 28 are 1, 2, 4, 7, 14, 28. Their HCF is 1.
45. What is the product of the HCF and LCM of 12 and 15?
A) 120
B) 150
C) 180
D) 200
Answer: C) 180
Explanation:
HCF = 3
LCM = 60
Product = 3 × 60 = 180, which equals 12 × 15.
46. What is the smallest number which when divided by 5, 6 and 7 leaves remainder 3?
A) 213
B) 210
C) 203
D) 183
Answer: A) 213
Explanation: LCM(5,6,7) = 210. Therefore, the required number is:
210 + 3 = 213.
47. A number is divisible by 6 if it is divisible by:
A) 2 and 3
B) 2 and 5
C) 3 and 5
D) 4 and 6
Answer: A) 2 and 3
Explanation: Since 6 = 2 × 3, a number must be divisible by both 2 and 3.
48. A number is divisible by 12 if it is divisible by:
A) 2 and 6
B) 3 and 4
C) 2 and 5
D) 4 and 6 only
Answer: B) 3 and 4
Explanation: Since 12 = 3 × 4 and 3 and 4 are co-prime, divisibility by both guarantees divisibility by 12.
49. Which of the following numbers is divisible by 4?
A) 1234
B) 1248
C) 1251
D) 1262
Answer: B) 1248
Explanation: For divisibility by 4, the last two digits must be divisible by 4. 48 is divisible by 4.
50. Which number is divisible by 8?
A) 1232
B) 1242
C) 1254
D) 1266
Answer: A) 1232
Explanation: The last three digits, 232, are divisible by 8:
232 ÷ 8 = 29.
51. What is the remainder when 12345 is divided by 5?
A) 0
B) 1
C) 2
D) 5
Answer: A) 0
Explanation: A number ending in 0 or 5 is divisible by 5. Hence, remainder = 0.
52. What is the remainder when 9876 is divided by 10?
A) 0
B) 4
C) 6
D) 8
Answer: C) 6
Explanation: Division by 10 leaves the last digit as the remainder. Therefore, remainder = 6.
53. What is the smallest number that should be added to 999 to make it divisible by 8?
A) 1
B) 5
C) 7
D) 9
Answer: A) 1
Explanation: 999 leaves remainder 7 when divided by 8. Adding 1 gives 1000, which is divisible by 8.
54. What is the smallest number that should be subtracted from 1000 to make it divisible by 9?
A) 1
B) 2
C) 3
D) 4
Answer: B) 1
Explanation: 1000 leaves remainder 1 when divided by 9. Therefore, subtract 1.
55. How many factors does 24 have?
A) 6
B) 7
C) 8
D) 9
Answer: C) 8
Explanation:
24 = 2³ × 3¹.
Number of factors = (3 + 1)(1 + 1) = 8.
56. How many factors does 36 have?
A) 7
B) 8
C) 9
D) 10
Answer: C) 9
Explanation:
36 = 2² × 3².
Number of factors = (2 + 1)(2 + 1) = 9.
57. How many factors does 100 have?
A) 7
B) 8
C) 9
D) 10
Answer: C) 9
Explanation:
100 = 2² × 5².
Number of factors = 3 × 3 = 9.
58. What is the smallest number having exactly three factors?
A) 2
B) 3
C) 4
D) 6
Answer: C) 4
Explanation: 4 has factors 1, 2 and 4. In general, a square of a prime has exactly three factors.
59. Which number has exactly two factors?
A) 1
B) 9
C) 13
D) 15
Answer: C) 13
Explanation: A number with exactly two positive factors is a prime number. 13 is prime.
60. What is the number of factors of 72?
A) 10
B) 12
C) 14
D) 16
Answer: B) 12
Explanation:
72 = 2³ × 3².
Number of factors = (3 + 1)(2 + 1) = 12.
61. What is √144?
A) 10
B) 11
C) 12
D) 14
Answer: C) 12
Explanation: 12 × 12 = 144.
62. What is √225?
A) 12
B) 15
C) 18
D) 20
Answer: B) 15
Explanation: 15² = 225.
63. Which of the following is a perfect square?
A) 45
B) 48
C) 49
D) 50
Answer: C) 49
Explanation: 49 = 7², so it is a perfect square.
64. Which of the following is a perfect cube?
A) 64
B) 72
C) 81
D) 100
Answer: A) 64
Explanation: 64 = 4³ = 4 × 4 × 4.
65. What is ∛125?
A) 3
B) 4
C) 5
D) 6
Answer: C) 5
Explanation: 5 × 5 × 5 = 125.
66. Which number lies between 10² and 11²?
A) 105
B) 110
C) 115
D) 121
Answer: C) 115
Explanation:
10² = 100
11² = 121
Therefore, 115 lies between them.
67. What is 15²?
A) 215
B) 225
C) 235
D) 245
Answer: B) 225
Explanation: 15 × 15 = 225.
68. What is 25²?
A) 525
B) 625
C) 725
D) 825
Answer: B) 625
Explanation: 25 × 25 = 625.
69. What is 99²?
A) 9701
B) 9801
C) 9901
D) 9999
Answer: B) 9801
Explanation:
99² = (100 − 1)²
= 10000 − 200 + 1
= 9801.
70. What is 101²?
A) 10101
B) 10201
C) 10301
D) 10401
Answer: B) 10201
Explanation:
101² = (100 + 1)²
= 10000 + 200 + 1
= 10201.
71. What is the value of 2⁰?
A) 0
B) 1
C) 2
D) Undefined
Answer: B) 1
Explanation: Any non-zero number raised to the power zero equals 1.
72. What is 5³?
A) 25
B) 75
C) 125
D) 150
Answer: C) 125
Explanation: 5³ = 5 × 5 × 5 = 125.
73. What is 10⁵?
A) 10,000
B) 100,000
C) 1,000,000
D) 1,000
Answer: B) 100,000
Explanation: 10⁵ means five factors of 10:
10 × 10 × 10 × 10 × 10 = 100,000.
74. What is the value of 2⁶?
A) 32
B) 64
C) 128
D) 256
Answer: B) 64
Explanation: 2⁶ = 2 × 2 × 2 × 2 × 2 × 2 = 64.
75. Which of the following is greater?
A) 2⁵
B) 3³
C) 4²
D) 5²
Answer: A) 2⁵.
Explanation:
2⁵ = 32
3³ = 27
4² = 16
5² = 25
Actually, the greatest is 32, so the correct answer is A) 2⁵.
76. What is the last digit of 3¹⁰?
A) 1
B) 3
C) 7
D) 9
Answer: D) 9.
Explanation: Unit digits of powers of 3 follow the cycle:
3, 9, 7, 1.
10 mod 4 = 2, so the unit digit corresponds to the second position: 9.
Therefore, the correct answer is D) 9.
77. What is the unit digit of 8⁴?
A) 2
B) 4
C) 6
D) 8
Answer: C) 6
Explanation: Unit digits of powers of 8 follow the cycle 8, 4, 2, 6. Since exponent 4 corresponds to the fourth position, the unit digit is 6.
78. What is the unit digit of 6²⁰²⁶?
A) 2
B) 4
C) 6
D) 8
Answer: C) 6
Explanation: Any positive power of a number ending in 6 also ends in 6.
79. What is the remainder when 10³ is divided by 7?
A) 1
B) 2
C) 3
D) 6
Answer: B) 6
Explanation: 1000 ÷ 7 = 142 remainder 6.
80. If a number is divisible by both 4 and 5, it must be divisible by:
A) 10
B) 15
C) 20
D) 25
Answer: C) 20
Explanation: Since 4 and 5 are co-prime, their LCM is 20. Therefore, the number must be divisible by 20.
81. What is the smallest number which leaves remainder 1 when divided by 2, 3 and 4?
A) 7
B) 9
C) 13
D) 25
Answer: C) 13.
Explanation: LCM(2,3,4) = 12. Numbers of the form 12k + 1 satisfy the condition. The smallest such number greater than the divisors is 13, not 7 because 7 ÷ 3 gives remainder 1, 7 ÷ 4 gives remainder 3. Therefore, the correct answer is C) 13.
82. What is the smallest number that is divisible by 9 and 12?
A) 18
B) 24
C) 36
D) 48
Answer: C) 36
Explanation:
9 = 3²
12 = 2² × 3
LCM = 2² × 3² = 36.
83. What is the HCF of 84 and 126?
A) 21
B) 28
C) 42
D) 63
Answer: C) 42
Explanation:
84 = 2² × 3 × 7
126 = 2 × 3² × 7
HCF = 2 × 3 × 7 = 42.
84. What is the LCM of 16 and 24?
A) 32
B) 40
C) 48
D) 64
Answer: C) 48
Explanation:
16 = 2⁴
24 = 2³ × 3
LCM = 2⁴ × 3 = 48.
85. The HCF of two consecutive natural numbers is always:
A) 0
B) 1
C) 2
D) Their sum
Answer: B) 1
Explanation: Consecutive numbers have no common factor greater than 1. Hence, their HCF is always 1.
86. What is the product of three consecutive natural numbers 4, 5 and 6?
A) 100
B) 110
C) 120
D) 130
Answer: C) 120
Explanation: 4 × 5 × 6 = 120.
87. The sum of two consecutive odd numbers is always:
A) Odd
B) Even
C) Prime
D) Zero
Answer: B) Even
Explanation: Odd + Odd = Even.
88. The product of two odd numbers is always:
A) Even
B) Odd
C) Zero
D) Prime
Answer: B) Odd
Explanation: An odd number has no factor 2. Therefore, multiplying two odd numbers produces an odd number.
89. The product of an even number and an odd number is always:
A) Odd
B) Even
C) Prime
D) Irrational
Answer: B) Even
Explanation: The even number contains factor 2, so the product will always contain factor 2.
90. If n is an even number, which expression is always even?
A) n + 1
B) n − 1
C) 2n
D) n/2
Answer: C) 2n
Explanation: Multiplying any integer by 2 always produces an even number.
91. What is the remainder when 99999 is divided by 9?
A) 0
B) 1
C) 8
D) 9
Answer: A) 0
Explanation: Digit sum = 9 + 9 + 9 + 9 + 9 = 45, which is divisible by 9.
92. What is the smallest number that must be multiplied by 12 to make it a perfect square?
A) 2
B) 3
C) 4
D) 6
Answer: B) 3
Explanation:
12 = 2² × 3.
To make all prime exponents even, multiply by 3:
12 × 3 = 36 = 6².
93. What is the smallest number that must be multiplied by 18 to make it a perfect square?
A) 2
B) 3
C) 6
D) 9
Answer: B) 2
Explanation:
18 = 2 × 3².
Multiplying by 2 gives:
18 × 2 = 36 = 6².
94. What is the smallest number that must be multiplied by 24 to make it a perfect cube?
A) 2
B) 3
C) 6
D) 9
Answer: C) 9
Explanation:
24 = 2³ × 3.
For a perfect cube, the exponent of 3 must become 3. We need two more 3s:
24 × 3² = 24 × 9 = 216 = 6³.
95. What is the smallest number that must be added to 50 to make it a perfect square?
A) 11
B) 12
C) 13
D) 14
Answer: C) 14
Explanation: The next perfect square after 50 is 64.
64 − 50 = 14.
96. What is the smallest number that must be subtracted from 100 to make it a perfect square?
A) 0
B) 1
C) 2
D) 4
Answer: A) 0
Explanation: 100 itself is a perfect square:
10² = 100.
97. What is the smallest number that must be added to 80 to make it divisible by 9?
A) 1
B) 7
C) 9
D) 10
Answer: B) 1
Explanation: 80 ÷ 9 leaves remainder 8. Adding 1 gives 81, which is divisible by 9.
Correct answer: A) 1.
98. If a number is divisible by 2 and 3, it is necessarily divisible by:
A) 5
B) 6
C) 8
D) 9
Answer: B) 6
Explanation: LCM(2,3) = 6, so a number divisible by both 2 and 3 is divisible by 6.
99. What is the greatest number that can divide 65 and 117 leaving the same remainder?
A) 13
B) 26
C) 39
D) 52
Answer: D) 52.
Explanation: If the same remainder is left, the divisor must divide their difference:
117 − 65 = 52.
The greatest divisor of 52 is 52, so mathematically the answer would be 52, since 65 ÷ 52 leaves 13 and 117 ÷ 52 leaves 13.
Therefore, the correct answer is D) 52.
100. If the sum of two numbers is 50 and their HCF is 5, which of the following could be their pair?
A) 10, 40
B) 15, 35
C) 20, 30
D) 25, 25
Answer: B) 15, 35
Explanation:
15 + 35 = 50, and HCF(15,35) = 5.
Therefore, the valid pair is 15 and 35.
Quick Revision: Number System Formulas
Important formulas
1. Sum of first n natural numbers
2. Sum of first n odd numbers
3. Sum of first n even numbers
4. For two positive integers
5. Number of factors
If
then number of factors is:
6. Divisibility rules
- 2: Last digit is even.
- 3: Sum of digits divisible by 3.
- 4: Last two digits divisible by 4.
- 5: Last digit is 0 or 5.
- 6: Divisible by both 2 and 3.
- 8: Last three digits divisible by 8.
- 9: Sum of digits divisible by 9.
- 10: Last digit is 0.
- 11: Difference between alternating digit sums is 0 or a multiple of 11.
Important exam topics covered
Natural & Whole Numbers • Integers • Rational & Irrational Numbers • Prime & Composite Numbers • HCF & LCM • Divisibility Rules • Remainders • Unit Digits • Factors • Perfect Squares • Perfect Cubes • Powers • Consecutive Numbers • Number Properties
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