AMC 8 2014 Problem 13
AMC8 2014 年第 13 题
奇偶性★★★☆☆
If and are integers and is even, which of the following is impossible?
若 和 都是整数，且 是偶数，那么下列哪种情况是不可能的？
- A.n and m are even / 和 都是偶数
- B.n and m are odd / 和 都是奇数
- C.n+m is even / n+m 是偶数
- D.n+m is odd / n+m 是奇数
- E.none of these are impossible / 以上都不是不可能的
正确答案 · Correct Answer
D
解析 · Solution
Solution 1
Since is even, either both and are even, or they are both odd. Therefore, and 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+m, , is odd.
We can assign values of n and m as 1,3. Then is 10. We can also assign values of n and m to be even, like 2 and 4. After calculating, we can determine that + could be odd or even. Then, we can conclude that n+m is not odd. Hence our answer choice is - TheNerdWhoIsNerdy.
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