Question

Hamlet 镇中，每一匹马对应 $3$ 个人；每一头牛对应 $4$ 只羊；每一个人对应 $3$ 只鸭。下列哪个数不可能是 Hamlet 镇中人、马、羊、牛、鸭的总数？

The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet?

$41$

$47$

$59$

$61$

$66$

答案：B

知识点：丢番图方程模运算

难度评级：1420

解答：

若有 $h$ 匹马和 $c$ 头牛，则有 $3h$ 个人、 $9h$ 只鸭和 $4c$ 只羊，总数为 $13h+5c$ 。

除 $47$ 外，其余选项都可表示为所需形式：对 $47$ ，减去 $0,13,26,39$ 后分别得到 $47,34,21,8$ ，没有一个是 $5$ 的倍数。 $\begin{gathered} 41=13\cdot2+5\cdot3, \\ 59=13\cdot3+5\cdot4, \\ 61=13\cdot2+5\cdot7, \\ 66=13\cdot2+5\cdot8. \end{gathered}$

所以正确答案是 B。

If there are $h$ horses and $c$ cows, then there are $3h$ people, $9h$ ducks, and $4c$ sheep. The total is therefore $13h+5c$ .

The listed values except $47$ can be written in that form: $\begin{gathered} 41=13\cdot2+5\cdot3, \\ 59=13\cdot3+5\cdot4, \\ 61=13\cdot2+5\cdot7, \\ 66=13\cdot2+5\cdot8. \end{gathered}$ For $47$ , subtracting $0,13,26,39$ leaves $47,34,21,8$ , none of which is divisible by $5$ .

Thus, the correct answer is B.

2015 AMC 10B 第 15 题

答案：B

难度评级：1420

解答：

其他年份的第 15 题