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 ?

Answer is (d). Consider following rules - 1. odd × odd = odd; 2. odd×even = even; 3. even × even = even; 4. odd + odd = even; 5. even + even = even and 6. even + odd = odd. By above rules p×q = even, if both p and q are even or they form a pair of odd and even numbers and p×q = odd, if p and q both are odd. Statement I: Information given is not sufficient as in a set of prime number two numbers can be even and odd or odd and odd. So, p×q can be even or odd. Statement II: Information given is not sufficient as it is not known that whether p and q are integers are not. They may be fractions. By combining statements I and II: we have p and q as prime number and there sum is odd. So, the two prime numbers should be one even and one odd. So now we have that one is even and one is odd. In this case p×q will always be even. So we can answer the question that No, p×q is not an odd integer.