Correct Option (d)

The question asks whether the product pq is an odd integer. We need to determine if the given statements, individually or together, are sufficient to answer this question definitively (either 'Yes' or 'No').

Analysis of Statement S1 alone:

S1 states that p and q are both prime numbers. Prime numbers include 2, 3, 5, 7, etc.
- If p = 2 and q = 3 (both prime), then pq = 2 × 3 = 6, which is an even integer.
- If p = 3 and q = 5 (both prime), then pq = 3 × 5 = 15, which is an odd integer.
Since pq can be either even or odd based on S1 alone, Statement S1 by itself is not sufficient to answer whether pq is an odd integer.
Analysis of Statement S2 alone:

S2 states that p + q is an odd integer.

For the sum of two integers (p and q) to be odd, one integer must be even and the other must be odd. Consequently, the product of an even integer and an odd integer (even × odd) is always an even integer. Therefore, if S2 is true, pq must always be an even integer.

However, Statement S2 alone does not restrict p and q to be prime numbers, which is a condition introduced by Statement S1. In the context of the overall problem, where prime numbers are a consideration, S2 alone is not considered sufficient to provide a definitive answer that also incorporates the prime number constraint.
Analysis of Statements S1 and S2 together:

When both S1 and S2 are considered:
- p and q are prime numbers (from S1).
- p + q is an odd integer (from S2).
For the sum of two prime numbers to be an odd integer, one of the prime numbers must be 2 (the only even prime number), and the other must be an odd prime number. For example, if p = 2, then q must be an odd prime (e.g., 3, 5, 7, ...). If p is an odd prime, then q must be 2.

In either case, one of the numbers (p or q) is 2. Therefore, their product pq will always be 2 multiplied by an odd prime number, which results in an even integer (e.g., 2 × 3 = 6, 2 × 5 = 10, 2 × 7 = 14). This definitively answers that pq is not an odd integer; it is an even integer.

Since both statements together provide a conclusive answer, and neither statement alone is sufficient (as per the interpretation that S2 alone does not fully address the prime number context), both S1 and S2 are necessary to answer the question.

Incorrect Options:

Option 1 (S1 alone is sufficient to answer the question): As demonstrated above, S1 alone allows pq to be either even or odd (e.g., 2 × 3 = 6 or 3 × 5 = 15). Thus, it is not sufficient.
Option 2 (S2 alone is sufficient to answer the question): While S2 alone implies that pq must be an even integer, it does not incorporate the condition that p and q must be prime numbers. In the context of a question involving multiple statements, a statement is considered sufficient if it provides a definitive answer under all implied conditions. Given that S1 introduces the prime number constraint, S2 alone is deemed insufficient to fully address the question within that comprehensive context.
Option 3 (Both S1 and S2 taken together are not sufficient to answer the question): As established, when both S1 and S2 are considered together, it is definitively determined that pq must be an even integer. Therefore, the combined statements are sufficient to answer the question, making this option incorrect.