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 ?
- (a) 4
- (b) 5 ✓ UPSC's answer
- (c) 6
- (d) 7
Why the answer is (b)
• If p and k were both odd primes, p² + k would be even and greater than 2, so it could not be prime. Hence p = 2 or k = 2.
• Case p = 2: 4 + k must be a prime below 30. k = 3, 7, 13, 19 give 7, 11, 17, 23 (all prime); k = 2, 5, 11, 17, 23 give 6, 9, 15, 21, 27 (not prime); k = 29 gives 33, which is above 30.
• Case k = 2 with p odd: p² + 2 < 30 allows p = 3 or 5; 3² + 2 = 11 is prime (5² + 2 = 27 is not). So k = 2 also works.
• Possible values of k: 2, 3, 7, 13, 19 — five values, option (b).
Why the other options are wrong
- (a) 4
- The count is 5, not 4, because k=2 is also a valid solution when p=3.
- (c) 6
- The count is 5, not 6, as no other prime k values satisfy the condition for p²+k being a prime less than 30.
- (d) 7
- The count is 5, not 7, as only five distinct prime values for k yield a prime sum less than 30.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2025, held on 25 May 2025. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.