We can solve this by using examples. Lets get down to statements.

Statement 1: Easiest numbers to take when a question is testing prime numbers and odd-even concept is to take 2 and 3.

If x=2 and y=6, then y/x=6/2=3 which is prime. Here, x is even

If x=3 and y=6, then y/x=6/3=2, which is prime. Here, x is odd.

Clearly, this statement is insufficient.Statement 2: For multiplication of two numbers to be prime, one of them has to necessarily be 1 and other has to be a prime number. We are already given x and y are positive integers. So, latter condition is satisfied. For the former, lets take the same cases we have taken before:

If x=2 and y=1, then xy=2 which is prime. Here, x is even

If x=3 and y=1, then xy=3, which is prime. Here, x is odd.

Clearly, this statement is insufficient.Note here: You could have taken x to be 1 in both cases, but that would really not give you a conclusive proof. Hence, we tested x to be a prime number.

Combining statements 1 & 2: The only way these two statements will be true is if x=1. This is because if x were any other positive integer, y will have to be a multiple of x, in order for statement 1 to give us a prime number. And if that was the case, xy would never be a prime number.

Let me take an example:

If x=2 and y=6, then y/x=6/2=3 which is prime. This satisfies statement 1

But, for statement 2, xy=2*6=12 which is not prime. Hence, statement 2 is not satisfied.

Thus, x has to be 1 which is odd.

Hence, correct answer is C
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