If A and B are positive integers…
We need to determine whether the product of positive integers AB is even. We must recall that the product of an even integer and any other integer is always even. Thus, if we can determine that either A or B (or both) is even, then we can answer the question.
Statement One Alone:
The sum A + B is odd.
In order for the sum of two integers to be odd, one of them must be even and the other must be odd. That is, either A is even and B is odd OR A is odd and B is even. In either case, the product AB is even, since one of them is even. Thus, statement one alone is sufficient to answer the question. We can eliminate answer choices B, C and E.
Statement Two Alone:
A is even.
Since we know that the product of an even integer and any other integer is always even, we know that the product AB will always be even.
Statement two is also sufficient.