A Question is given followed by two Statements I and II. Consider the Question and the Statements. There are three distinct prime numbers whose sum is a prime number. Question : What are those three numbers? Statement-I : Their sum is less than 23. Statement-II : One of the numbers is 5. Which one of the following is correct in respect of the above Question and the Statements?

(a) The Question can be answered by using one of the Statements alone, but cannot be answered using the other Statement alone ✓ UPSC's answer
(b) The Question can be answered by using either Statement alone
(c) The Question can be answered by using both the Statements together, but cannot be answered using either Statement alone
(d) The Question cannot be answered even by using both the Statements together

Why the answer is (a)

• If 2 were one of the three primes, the sum would be 2 + odd + odd = even and greater than 2 — not prime. So all three primes are odd.

• Statement I (sum < 23): the odd triples are {3, 5, 7} = 15, {3, 5, 11} = 19, {3, 5, 13} = 21, {3, 7, 11} = 21 (others reach 23 or more). Only 19 is prime, so the numbers are 3, 5, 11 — Statement I alone answers the question.

• Statement II (one number is 5): {3, 5, 11} = 19, {5, 7, 11} = 23, {5, 11, 13} = 29 and more all have prime sums — not unique.

• One statement alone is enough, the other is not — option (a).

Why the other options are wrong

(b) The Question can be answered by using either Statement alone: Statement II alone allows multiple valid sets such as {3,5,11}, {5,7,11}, and {5,11,13}, so it cannot be answered using either statement alone.
(c) The Question can be answered by using both the Statements together, but cannot be…: Statement I alone is sufficient to uniquely identify the set {3,5,11}, so the condition that it cannot be answered using either statement alone is false.
(d) The Question cannot be answered even by using both the Statements together: The question can be answered using Statement I alone, so it is incorrect to say it cannot be answered even with both statements.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.