AMC8 2014

AMC8 2014 · Q13

AMC8 2014 · Q13. It mainly tests Parity (odd/even).

If $n$ and $m$ are integers and $n^2+m^2$ is even, which of the following is impossible?

如果 $n$ 和 $m$ 是整数，且 $n^2+m^2$ 是偶数，下列哪项是不可能的？

(A) $n$ and $m$ are even $n$ and $m$ are even

(B) $n$ and $m$ are odd $n$ and $m$ are odd

(D) $n+m$ is odd $n+m$ is odd

(E) none of these are impossible none of these are impossible

Answer

Correct choice: (D)

正确答案：（D）

Solution

Since $n^2+m^2$ is even, either both $n^2$ and $m^2$ are even, or they are both odd. Therefore, $n$ and $m$ are either both even or both odd, since the square of an even number is even and the square of an odd number is odd. As a result, $n+m$ must be even. The answer, then, is $n^2+m^2$ $\boxed{(\text{D})}$ is odd.

由于 $n^2+m^2$ 是偶数，要么 $n^2$ 和 $m^2$ 都是偶数，要么它们都为奇数。因此，$n$ 和 $m$ 要么都是偶数，要么都是奇数，因为偶数的平方是偶数，奇数的平方是奇数。因此，$n+m$ 必须是偶数。答案是 $n^2+m^2$ 为奇数 $\boxed{(\text{D})}$ 是不可能的。

Topics

NT.005 Parity (odd/even)