Consider two statements S1 and S2 followed by a question : S1 : p and q both are prime numbers. S2 : p + q is an odd integer. Question : Is pq an odd integer? Which one of the following is correct ?

(a) S1 alone is sufficient to answer the question
(b) S2 alone is sufficient to answer the question ✓ UPSC's answer
(c) Both S1 and S2 taken together are not sufficient to answer the question
(d) Both S1 and S2 are necessary to answer the question

Why the answer is (b)

• S2 states that p + q is an odd integer. For the sum of two integers to be odd, exactly one of them must be even and the other must be odd.

• If one of p, q is even and the other is odd, then their product pq = (even) × (odd) is necessarily even.

• Therefore, from S2 alone we can conclude with certainty that pq is NOT an odd integer; the answer to the question is a definitive "No."

• S1 alone is not sufficient: if p = 2 and q = 3, then pq = 6 (even), but if p = 3 and q = 5, then pq = 15 (odd), so S1 by itself leaves the answer undetermined.

• Since S2 alone already gives a conclusive answer, S1 is not required, and the correct choice is (b).

Why the other options are wrong

(a) S1 alone is sufficient to answer the question: S1 alone does not determine the answer because p = 2, q = 3 gives pq = 6 (even) while p = 3, q = 5 gives pq = 15 (odd); the product can be either even or odd under S1 alone.
(c) Both S1 and S2 taken together are not sufficient to answer the question: S2 alone is already sufficient to conclude that pq is even, so the two statements taken together are certainly sufficient; claiming they are not sufficient is incorrect.
(d) Both S1 and S2 are necessary to answer the question: S1 is not necessary because S2 by itself forces one of p, q to be even, making pq even regardless of whether p and q are prime; requiring both statements is an overstatement.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2019, held on 2 June 2019. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.