GMAT Data Insights Question of the Day

1. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
2. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient. (correct answer)
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient to answer the question.

Explanation: This data sufficiency question tests your understanding of even/odd properties and how they interact through different operations. The key insight is recognizing what each statement tells you about the parity (even/odd nature) of $x$ and $y$. For $xy$ to be even, at least one of $x$ or $y$ must be even. If both are odd, their product is odd. Statement (1) tells us $x + y$ is odd. This happens only when one number is even and the other is odd (since even + even = even, and odd + odd = even). If exactly one of $x$ or $y$ is even, then their product $xy$ must be even. Statement (1) alone is sufficient. Statement (2) tells us $x^2 + y^2$ is even. Since squaring preserves parity (even² = even, odd² = odd), this means we have either even² + even² = even or odd² + odd² = even. Both scenarios give an even sum, so we can't determine whether $x$ and $y$ have the same parity or different parities. For example, if $x = 1, y = 3$, then $x^2 + y^2 = 10$ (even) but $xy = 3$ (odd). If $x = 1, y = 2$, then $x^2 + y^2 = 5$ (odd), contradicting our statement. Actually, if $x = 2, y = 4$, then $x^2 + y^2 = 20$ (even) and $xy = 8$ (even). Statement (2) alone is insufficient. Answer choice B is correct because statement (1) alone is sufficient while statement (2) alone is not. Choices A and D incorrectly assess statement (2) as sufficient. Choice C incorrectly claims both statements are needed. Strategy tip: For even/odd questions, systematically test what each parity combination produces, and remember that addition and multiplication have different parity rules.