Track / Stage 4 / 242 of 340 #502 of 2000

Problem 502

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

13. If $m^{2}+n^{2}(m, n \in \mathbf{Z})$ is even, then which of the following statements ( ) cannot be correct.
(A) $m, n$ are both even
(B) $m, n$ are both odd
(C) $m+n$ is even
(D) $m+n$ is odd
(E) One of the above answers is impossible

Your answer

Official solution

13. D.

From the problem, we know that $m^{2}$ and $n^{2}$ have the same parity.
When $m^{2}$ and $n^{2}$ are both even, $m$ and $n$ are both even, so $m+n$ is even;

When $m^{2}$ and $n^{2}$ are both odd, $m$ and $n$ are both odd, so $m+n$ is even.
Therefore, it is impossible for $m+n$ to be odd.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.