Question

下列哪个整数不能写成四个连续奇整数之和？

Which of the following integers cannot be written as the sum of four consecutive odd integers?

$16$

$40$

$72$

$100$

$200$

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设四个连续奇整数为 $2k+1,2k+3,2k+5,2k+7$ 。它们的和为 $8k+16=8(k+2).$

所以这样的和必须是 $8$ 的倍数。选项中唯一不能被 $8$ 整除的是 $100$ 。

所以正确答案是 D。

Let the four consecutive odd integers be $2k+1,2k+3,2k+5,2k+7$ . Their sum is $8k+16=8(k+2).$

So any such sum must be a multiple of $8$ . The only answer choice that is not divisible by $8$ is $100$ .

Thus, D is the correct answer.