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, which implies that one of the numbers is even and the other is odd, as the sum of two odd or two even numbers is always even.

• Since p and q are prime numbers (from S1), the only even prime number is 2.

• Therefore, one of the primes must be 2, and the other must be an odd prime.

• The product of an even number (2) and an odd number is always even.

• Thus, pq is an even integer, not an odd integer, so the answer to the question is 'No'.

• Since S2 alone determines the parity of the sum, which in turn determines the parity of the product for primes, S2 alone is sufficient to answer the question.

Why the other options are wrong

(a) S1 alone is sufficient to answer the question: S1 alone is insufficient because p and q could both be odd primes (e.g., 3 and 5), making pq odd, or one could be 2, making pq even.
(c) Both S1 and S2 taken together are not sufficient to answer the question: Both statements together are sufficient because S2 restricts the parity of the sum, allowing us to deduce that one prime is 2, thus determining the product is even.
(d) Both S1 and S2 are necessary to answer the question: S1 is not necessary to answer the question because S2 alone implies that one number is even and the other odd; however, without S1, we cannot assume they are primes, but in the context of the question asking about primes, S2's constraint on the sum is the decisive factor for the parity logic provided the numbers are integers, but strictly speaking, S2 alone tells us one is even and one is odd, so

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.