AMC8 2025

AMC8 2025 · Q6

AMC8 2025 · Q6. It mainly tests Remainders & modular arithmetic, Digit properties (sum of digits, divisibility tests).

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

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

(A) \ 15 \ 15

(B) \ 16 \ 16

(D) \ 18 \ 18

(E) \ 19 \ 19

Answer

Correct choice: (C)

正确答案：（C）

Solution

Notice the $10$'s place as you erase one number; you get $40$, which is a multiple of $4$, so you can ignore the $10$'s place in the numbers provided altogether because it will always be a multiple of 40. This means that the only relevant numbers are the $1$'s digit ones. If you add the numbers now, you get $5+6+7+8+9=35.$ The closest multiple of $4$ underneath $35$ which can be achieved as a result of subtracting either $5$, $6$, $7$, $8$, or $9$ is $28$, which is achieved by subtracting $7$, therefore the answer is $\boxed{\textbf{(C)}~17}$.

注意擦掉一个数字时 $10$ 位的影响；你得到 $40$，它是 $4$ 的倍数，所以可以完全忽略这些数字的 $10$ 位，因为它总是 $40$ 的倍数。这意味着唯一相关的数字是个位数。现在将它们相加，得到 $5+6+7+8+9=35$。$35$ 下方最近的 $4$ 的倍数是通过减去 $7$ 得到的 $28$，因此答案是 $\boxed{\textbf{(C)}~17}$。

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