Is the sum of two positive integers, x + y an odd value?

For what it is worth, I will add my own thoughts on how I solved the question. First, the stem provides a vital piece of information: x and y are both positive integers: not 0, not anything negative, no decimals. Since we are asked whether their sum is odd, we can think of possible valid combinations:

1) O + E = O
2) E + O = O

Of course, E + E = E and O + O = E, so we really need to find out whether the statements, either alone or together, can tell us that one unknown is odd and the other even.

Statement (1)

Think of the first few prime numbers: 2, 3, 5, 7, 11. Good enough. An important consideration here is that 2 is the only even prime number. If one of the unknowns is 2 and the other anything else, the sum will be odd. Otherwise, the sum will be even, even if both unknowns are 2. (Be careful: the problem does not stipulate that the integers must be unique.) The possibilities:

1) 2 + O = O
2) O + O = E
3) 2 + 2 = E

Hence, Statement (1) is NOT SUFFICIENT.

Statement (2)

The phrasing is a little odd-sounding, but successor should not be hard to figure out: it is simply the integer after each unknown. Since we already looked at the first few prime numbers above, we can appreciate the gap between 7 and 11, specifically 8 and 9, whose successors would be the non-primes 9 and 10, respectively. Again, organize the information:

1) 8 + 9 = O
2) 8 + 8 = E
3) 9 + 9 = E

The exact values of the two unknowns hardly matter. What is important is that we have found valid inputs for each, according to the statement, and come up with different results. Thus, Statement (2) is NOT SUFFICIENT.

Together

The statements tell us that 2 cannot be the value of either unknown, since the integer after 2 is 3, a prime number. Because all other prime numbers are odd, we can determine that the sum, regardless of the exact values of x and y, would consist of the following:

1) O + O = E

Even though the answer to the question is no, it is a definitive answer, so the answer to the question is (C). This is a good exercise in number sense and in keeping information organized.

Good luck with your studies.

- Andrew