Q.1 What is the greatest common divisor (GCD) of 24 and 36?
6
12
18
24
Explanation - The divisors of 24 are {1,2,3,4,6,8,12,24} and of 36 are {1,2,3,4,6,9,12,18,36}. The largest common divisor is 12.
Correct answer is: 12
Q.2 Which of the following is a prime number?
21
29
35
39
Explanation - A prime number has only two divisors: 1 and itself. 29 satisfies this condition.
Correct answer is: 29
Q.3 What is Euler’s totient φ(9)?
3
4
6
9
Explanation - φ(9) counts integers less than 9 that are coprime to 9. These are {1,2,4,5,7,8}, so φ(9)=6.
Correct answer is: 6
Q.4 The least common multiple (LCM) of 15 and 20 is?
30
60
90
120
Explanation - LCM(15,20) = (15×20)/GCD(15,20) = 300/5 = 60.
Correct answer is: 60
Q.5 What is the remainder when 2^10 is divided by 7?
1
2
3
4
Explanation - 2^10 = 1024. Dividing 1024 by 7 gives remainder 2.
Correct answer is: 2
Q.6 Which number is NOT divisible by 11?
121
242
330
451
Explanation - Divisibility test: difference of sum of alternating digits. For 451: (4+1)-5=0 → divisible? Actually remainder exists, so not divisible.
Correct answer is: 451
Q.7 What is the smallest prime factor of 91?
7
11
13
17
Explanation - 91 ÷ 7 = 13, hence 7 is the smallest prime factor.
Correct answer is: 7
Q.8 Which of the following is a perfect square?
225
245
255
265
Explanation - 225 = 15^2, a perfect square.
Correct answer is: 225
Q.9 What is the remainder when 100 is divided by 9?
0
1
2
4
Explanation - 100 ÷ 9 = 11 remainder 1.
Correct answer is: 1
Q.10 The sum of the first 10 natural numbers is?
45
50
55
60
Explanation - Sum of first n natural numbers = n(n+1)/2 = 10×11/2 = 55.
Correct answer is: 55
Q.11 Which of these numbers is a Carmichael number?
561
221
77
91
Explanation - 561 is the smallest Carmichael number (composite but satisfies Fermat’s test).
Correct answer is: 561
Q.12 What is the unit digit of 7^202?
1
3
7
9
Explanation - Units digit cycle of powers of 7: (7,9,3,1). 202 mod 4 = 2 → units digit = 9.
Correct answer is: 9
Q.13 The modular inverse of 3 modulo 11 is?
3
4
7
8
Explanation - 3×4 = 12 ≡ 1 (mod 11). Hence inverse of 3 mod 11 is 4.
Correct answer is: 4
Q.14 How many positive divisors does 60 have?
10
12
14
16
Explanation - Prime factorization 60 = 2^2 × 3^1 × 5^1. Number of divisors = (2+1)(1+1)(1+1)=12.
Correct answer is: 12
Q.15 Which is the smallest prime greater than 100?
101
103
107
109
Explanation - 101 is prime and the first prime after 100.
Correct answer is: 101
Q.16 If p is prime, which of the following must always hold?
p divides (p-1)!
(p-1)! ≡ -1 (mod p)
p divides (p+1)!
(p+1)! ≡ -1 (mod p)
Explanation - This is Wilson’s Theorem.
Correct answer is: (p-1)! ≡ -1 (mod p)
Q.17 Which is the binary representation of decimal 25?
11001
10101
10011
11100
Explanation - 25 = 16+8+1 → binary is 11001.
Correct answer is: 11001
Q.18 What is the digit sum of 987654?
39
42
45
48
Explanation - 9+8+7+6+5+4 = 39.
Correct answer is: 39
Q.19 Which number is relatively prime to 20?
12
14
21
25
Explanation - GCD(20,21)=1, so 21 is coprime with 20.
Correct answer is: 21
Q.20 What is the prime factorization of 84?
2^2×3×7
2×3×14
2^3×7
3^2×7×2
Explanation - 84 = 2×42 = 2×2×21 = 2^2×3×7.
Correct answer is: 2^2×3×7
Q.21 Which of the following is a Pythagorean triple?
6,8,10
7,8,9
10,12,14
15,20,26
Explanation - 6^2+8^2=36+64=100=10^2.
Correct answer is: 6,8,10
Q.22 What is 2^16 modulo 5?
1
2
3
4
Explanation - 2^4 ≡ 1 (mod 5). So 2^16=(2^4)^4 ≡1^4=1 (mod 5).
Correct answer is: 1
Q.23 What is the sum of prime numbers less than 10?
17
18
20
21
Explanation - Primes less than 10 are 2,3,5,7. Sum=17.
Correct answer is: 17
Q.24 The Fermat little theorem states that if p is prime and a is not divisible by p, then:
a^(p-1) ≡ 1 (mod p)
a^(p-1) ≡ 0 (mod p)
a^p ≡ 1 (mod p)
a^p ≡ 0 (mod p)
Explanation - This is Fermat’s little theorem.
Correct answer is: a^(p-1) ≡ 1 (mod p)
Q.25 What is the largest two-digit prime number?
89
97
99
101
Explanation - 97 is prime and the largest prime below 100.
Correct answer is: 97