Let both p and k be prime numbers such that (p² + k) is also a prime number less than 30. What is the number of possible values of k?
Answer & explanation
Answer: (b) 5
If p and k were both odd, p² + k would be even and larger than 2, so one of them must be 2. With p = 2, k can be 3, 7, 13 or 19; with k = 2 (and p = 3, giving 11) k = 2 also works, so k has 5 possible values.
- If p and k are both odd primes, p² + k is odd + odd = even and greater than 2, so it cannot be prime. Hence p = 2 or k = 2.
- p = 2: 4 + k must be a prime below 30. k = 3, 7, 13, 19 give 7, 11, 17, 23 (prime); k = 2, 5, 11, 17, 23 give 6, 9, 15, 21, 27 (not prime).
- k = 2: p² + 2 must be a prime below 30. p = 3 gives 11 (prime); p = 5 gives 27 (not prime). So k = 2 is possible.
- Possible values of k: 2, 3, 7, 13, 19 — five values.
Remember · Odd + odd is even: when a sum of primes must be prime, one of them is usually 2.
Question and answer: UPSC's official GS Paper II (2025, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·