2006 AMC 8 第 23 题

Question

一个盒子里有金币。如果把金币平均分给六个人，会剩下四枚。如果平均分给五个人，会剩下三枚。如果盒子中装的是满足这两个条件的最少金币数，那么平均分给七个人时会剩下几枚？

A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?

0

1

2

3

5

答案：A

知识点：模运算系统列举

难度评级：1450

小提示：

列出除以

6

余

4

的较小正整数。

List small numbers that leave remainder

4

when divided by

6

.

大提示：

等价地，加两枚金币后应同时能被

5

和

6

整除。

Equivalently, adding two coins would make the number divisible by both

5

and

6

.

解答：

除以

6

余

4

的正整数为

4, 10, 16, 22, 28, 34, 40, \cdots\text{。}

除以

5

余

3

的正整数为

3, 8, 13, 18, 23, 28, 33, \cdots\text{。}

可见满足条件的最少金币数是

28

。它除以

7

余

0

。

所以正确答案是 A。

The positive integers that leave a remainder of

4

when divided by

6

are

4, 10, 16, 22, 28, 34, 40, \cdots.

The positive integers that leave a remainder of

3

when divided by

5

are

3, 8, 13, 18, 23, 28, 33, \cdots.

From this, we can see that the smallest number of coins that work is

28.

This leaves a remainder of

0

when divided by

7.

Thus, A is the correct answer.

2006 AMC 8 第 23 题

答案：A

难度评级：1450

小提示：

大提示：

解答：

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