AMC10 2015 B

AMC10 2015 B · Q15

AMC10 2015 B · Q15. It mainly tests Ratios & proportions, GCD & LCM.

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?

哈姆雷特镇每匹马有3个人，每头牛有4只羊，每个人有3只鸭。以下哪个不可能是哈姆雷特镇的总人数、马、羊、牛和鸭的数量？

(A) 41 41

(B) 47 47

(D) 61 61

(E) 66 66

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Let $h$ be the number of horses and $c$ be the number of cows. There are then $3h$ people, $9h$ ducks, and $4c$ sheep in Hamlet. The total population of Hamlet is $13h+5c$, where $h$ and $c$ are whole numbers. A number $N$ can be the population only if there exists a whole number value for $h$ such that $N-13h$ is a whole number multiple of $5$. This is possible for all the provided numbers except $47$, as follows: $41-13\cdot 2=5\cdot 3$, $59-13\cdot 3=5\cdot 4$, $61-13\cdot 2=5\cdot 7$, and $66-13\cdot 2=5\cdot 8$. None of $47$, $47-13=34$, $47-13\cdot 2=21$, and $47-13\cdot 3=8$ is a multiple of $5$. Therefore $47$ cannot be the population of Hamlet. Note: In fact, $47$ is the largest number that cannot be the population.

答案（B）：设 $h$ 为马的数量，$c$ 为牛的数量。那么哈姆雷特村里有 $3h$ 个人、$9h$ 只鸭子，以及 $4c$ 只羊。哈姆雷特村的总人口为 $13h+5c$，其中 $h$ 和 $c$ 都是整数。一个数 $N$ 能成为人口数，当且仅当存在某个整数 $h$ 使得 $N-13h$ 是 $5$ 的整数倍。除 $47$ 外，题目给出的所有数都满足这一条件，例如：$41-13\cdot 2=5\cdot 3$，$59-13\cdot 3=5\cdot 4$，$61-13\cdot 2=5\cdot 7$，以及 $66-13\cdot 2=5\cdot 8$。而 $47$、$47-13=34$、$47-13\cdot 2=21$、$47-13\cdot 3=8$ 都不是 $5$ 的倍数。因此 $47$ 不可能是哈姆雷特村的人口数。注：事实上，$47$ 是不能成为人口数的最大整数。

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