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 ?

To determine if the product pq is an odd integer, we must analyze the statements provided. Statement S1 tells us p and q are both prime numbers. If p and q are 3 and 5, their product 15 is odd. However, if one of them is 2 which is the only even prime, their product would be even. Therefore, S1 alone is not sufficient because the product could be odd or even. Statement S2 tells us p + q is an odd integer. For the sum of two integers to be odd, one must be even and the other must be odd. This means one of the numbers is even. However, S2 does not specify if they are prime numbers or any other type of integers. Without knowing if the odd number is an integer or a fraction, we cannot definitively conclude the nature of the product. When we take both statements together, S2 confirms that one number is even and the other is odd. S1 tells us both are prime. Since 2 is the only even prime number, one of the numbers must be 2. Since 2 is even, the product of 2 and any other prime number will always be even. Therefore, we can now answer the question Is pq an odd integer with a definite No. Both statements are necessary to reach this conclusion. This makes Option D the correct choice.