Question

设 $N$ 是唯一的正整数，使得 $273436$ 除以 $N$ 的余数为 $16$ ，并且 $272760$ 除以 $N$ 的余数为 $15$ 。 $N$ 的十位数字是多少？

Let $N$ be the unique positive integer such that dividing $273436$ by $N$ leaves a remainder of $16,$ and dividing $272760$ by $N$ leaves a remainder of $15.$ What is the tens digit of $N?$

$0$

$1$

$2$

$3$

$4$

答案：E

知识点：最大公约数整除性

难度评级：1730

解答：

减去余数。 $273420 = 273436 - 16$ 和 $272745 = 272760 - 15$ 都是 $N$ 的倍数，所以它们的差 $675$ 也是倍数。又有 $272745 = 404 \cdot 675 + 45$ ，因此 $45$ 也是 $N$ 的倍数。余数 $16$ 意味着 $N \gt 16$ ，而 $45$ 的大于 $16$ 的因数只有 $45$ 本身。所以 $N = 45$ ，十位数字是 $4$ 。因此正确答案是 E。

Subtract off the remainders. Both $273420 = 273436 - 16$ and $272745 = 272760 - 15$ are multiples of $N,$ so their difference $675$ is too. Now $272745 = 404 \cdot 675 + 45,$ which makes $45$ a multiple of $N$ as well. The remainder $16$ means $N \gt 16,$ and the only divisor of $45$ bigger than $16$ is $45$ itself. So $N = 45,$ and its tens digit is $4.$ Thus, E is the correct answer.

2025 AMC 10A 第 17 题

答案：E

难度评级：1730

解答：

其他年份的第 17 题