Consider two statements S1 arid 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?

The correct option is S2 alone is sufficient to answer the question.

Let's analyze the question and the statements step-by-step to understand why statement S2 alone is sufficient to answer the question, while S1 is not.

The question asks: Is $pq$ an odd integer?

For the product of two integers $p$ and $q$ (i.e., $pq$) to be an odd integer, both $p$ and $q$ must be odd integers. If either of them (or both) is even, their product $pq$ will be an even integer.

Analyzing Statement S1: "p and q both are prime numbers."
If $p$ and $q$ are prime numbers, they can be odd or even. The only even prime number is 2, and all other prime numbers are odd.
- Case 1: If $p=3$ and $q=5$ (both are prime), then $pq=3\times 5=15$, which is an odd integer (Yes).
- Case 2: If $p=2$ and $q=3$ (both are prime), then $pq=2\times 3=6$, which is an even integer (No).
Since we get two different answers ("Yes" and "No"), S1 alone is not sufficient to answer the question.

Analyzing Statement S2: "p + q is an odd integer."
The sum of two integers is odd if and only if one of the integers is even and the other is odd.
Therefore, statement S2 directly implies that one of $p$ or $q$ must be an even integer, and the other must be an odd integer.
Since one of the numbers is even, their product $pq$ must be an even integer.
Consequently, $pq$ cannot be an odd integer, giving us a definite answer of "No".
Since we get a unique and definite answer ("No"), S2 alone is sufficient to answer the question.