If m and n are different positive integers, then how many prime numbers are in set {m, n, m + n}? (1) mn is prime. (2) m + n is even.
A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not. B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not. C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient. D. Either statement BY ITSELF is sufficient to answer the question. E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.
(C) Statement (1) tells us that mn is prime. The product of two integers can be prime only if one integer is 1 and the other one is prime. Otherwise the product would be divisible by more factors than just itself and 1. We can plug in some small prime numbers to see that statement (1) by itself is NOT sufficient. The set can be {1, 2, 3} – 2 prime numbers, or it can be {1, 3, 4} – 1 prime number.
Statement (2) tells us that m + n is even. Therefore the integers are both odd or both even. Clearly, there are many options possible: {2, 4, 6} – 1 prime number, {4, 6, 8} – no prime numbers, {3, 5, 8} – 2 prime numbers. Therefore statement (2) by itself is NOT sufficient.
If we use the both statements together, the set is {1, n, n + 1}, where n is prime and odd. If n is odd, then n + 1 is even. So n + 1 is NOT prime. Therefore the set contains exactly one prime number. Statements (1) and (2) taken together are sufficient to answer the question, even though neither statement by itself is sufficient. The correct answer is C.  In case of {1, n, n + 1}, what if n = 2, then n + 1 = 3. This makes the set as {1, 2, 3}. Thus 2 prime numbers. Hence this too is insufficient. Hence, the answer should be E.
