Theory of Numbers

JKPSC Assistant Professor (Mathematics) · Mathematics · 49 practice questions with answers and explanations: 5 Easy, 40 Medium, 4 Hard.

Sample questions

Sample Easy question 1

Which of these is a prime number?

  1. 27
  2. 29
  3. 39
  4. 33
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Answer: B. The others are divisible by 3.

Sample Easy question 2

The least common multiple of 4 and 6 is:

  1. 10
  2. 24
  3. 12
  4. 2
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Answer: C. The lcm is 2² × 3.

Sample Easy question 3

The sum of the first n odd positive integers is:

  1. n(n + 1)
  2. 2n
  3. n²
  4. n³
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Answer: C. 1 + 3 + 5 + … + (2n − 1) = n².

Sample Medium question 4

The Möbius function μ(n) is zero when n:

  1. Has a squared prime factor
  2. Is prime
  3. Is odd
  4. Is a perfect square only
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Answer: A. μ(n) is non-zero only for square-free n.

Sample Medium question 5

The value of μ(30) is:

  1. 0
  2. −1
  3. 3
  4. 1
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Answer: B. 30 = 2 × 3 × 5 has three distinct prime factors, so μ = (−1)³.

Sample Medium question 6

Euclid proved that:

  1. There are finitely many primes
  2. Every even number is a sum of two primes
  3. There are infinitely many primes
  4. There are infinitely many twin primes
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Answer: C. His proof is by contradiction.

Sample Hard question 7

By Wilson's theorem, 6! mod 7 equals:

  1. 5
  2. 6
  3. 1
  4. 0
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Answer: B. 6! ≡ −1 ≡ 6 (mod 7).

Sample Hard question 8

The remainder when 7^100 is divided by 5 is:

  1. 3
  2. 2
  3. 1
  4. 4
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Answer: C. 7 ≡ 2 (mod 5), and 2⁴ ≡ 1, so 2^100 = (2⁴)^25 ≡ 1.

Sample Hard question 9

The number of integers from 1 to 100 that are coprime to 100 is:

  1. 50
  2. 99
  3. 20
  4. 40
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Answer: D. φ(100) = 100 × (1 − 1/2) × (1 − 1/5) = 40.

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