Question

Sekou 写下 $15$ 、 $16$ 、 $17$ 、 $18$ 、 $19$ 。擦掉其中一个数后，剩下四个数的和是 $4$ 的倍数。他擦掉了哪个数？

Sekou writes the numbers $15,$ $16,$ $17,$ $18,$ $19.$ After he erases one of the numbers, the sum of the remaining four numbers is a multiple of $4.$ Which number did he erase?

$15$

$16$

$17$

$18$

$19$

答案：C

知识点：模运算整除性

难度评级：900

小提示：

比较擦掉前后的总和模 $4$ 的余数

Compare the total sum modulo $4$ before and after erasing

大提示：

被擦掉的数必须与原总和有相同的模四余数

The erased number must have the same remainder as the original total

视频讲解：

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文字解答：

这五个数除以 $4$ 的余数分别为 $3$ 、 $0$ 、 $1$ 、 $2$ 、 $3$ 。因此五个数的总和模 $4$ 与 $3 + 0 + 1 + 2 + 3 = 9$ 相同，余数为 $1$ （模 $4$ ）。要擦掉一个数后使总和余 $0$ （模 $4$ ），必须擦掉一个余数为 $1$ （模 $4$ ）的数，也就是 $17$ 。所以答案是 C。

The remainders of the five numbers after dividing by $4$ are: $3,$ $0,$ $1,$ $2,$ and $3.$ So, the sum of all five numbers modulo $4$ is the same as $3 + 0 + 1 + 2 + 3 = 9$ which has remainder $1$ modulo $4.$ Therefore, in order to erase a single number and get a sum that is $0$ modulo $4,$ we must erase the number which was $1$ modulo $4,$ which was $17.$ Therefore, the answer is C.

2025 AMC 8 第 6 题