Question

Boris 有一台神奇的换硬币机器。他放入一枚二十五分硬币，机器会吐出五枚五分硬币；放入一枚五分硬币，会吐出五枚一分硬币；放入一枚一分硬币，会吐出五枚二十五分硬币。Boris 一开始只有一枚一分硬币。反复使用这台机器后，他可能拥有下列哪一个金额？

Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?

$3.63

$5.13

$6.30

$7.45

$9.07

答案：D

知识点：不变量模运算钱币

难度评级：1750

解答：

二十五分硬币换五分硬币，或五分硬币换一分硬币，都保持总金额不变。只有一分硬币换五枚二十五分硬币时，金额增加 $5 \cdot 25 - 1 = 124$ 美分。

从 $1$ 美分开始，Boris 的总金额总是 $1 + 124n$ 美分，其中 $n$ 是非负整数。

只有 $\$7.45$ 符合，因为 $745 = 1 + 124 \cdot 6$ 。

所以正确答案是 D。

Trading a quarter for five nickels or a nickel for five pennies does not change the total value. Only trading a penny for five quarters changes it, adding $5 \cdot 25 - 1 = 124$ cents.

Starting from $1$ cent, Boris always has $1 + 124n$ cents for some nonnegative integer $n.$

Only $\$7.45$ has this form, since $745 = 1 + 124 \cdot 6.$

Thus, the correct answer is D.

2000 AMC 10 第 17 题

答案：D

难度评级：1750

解答：

其他年份的第 17 题