AMC 8 1986 Problem 17
AMC8 1986 年第 17 题
奇偶性★★★☆☆
Let be an odd whole number and let be any whole number. Which of the following statements about the whole number is always true?
设 为一个奇整数， 为任意整数。关于整数 ，以下哪条陈述一定成立？
- A. / 它总是奇数
- B. / 它总是偶数
- C. / 仅当 为偶数时它才是偶数
- D. / 仅当 为奇数时它才是奇数
- E. / 仅当 为偶数时它才是奇数
正确答案 · Correct Answer
E
解析 · Solution
Solution 1
Since is odd, must be odd.
If is odd, then $no$ is odd, so is even.
If is even, then $no$ is even, so is odd.
Therefore, of the answer choices, only accurately describes .
We are given that , so in mod 2 we have which is odd only if is even
If o = 1 and n = 2, then , which eliminates (B) and (D). If o = 1 and n = 1, then , which eliminates (A) and (C). This leaves as the correct answer.
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