2020 AMC 10A 第 9 题

Question

学校活动中，一个长凳单元可以坐

7

名成人或

11

名儿童。当

N

个长凳单元首尾相连时，相同人数的成人和儿童一起就坐，刚好占满所有长凳空间。

N

的最小正整数值是多少？

A single bench section at a school event can hold either

7

adults or

11

children. When

N

bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of

N?

9

18

27

36

77

答案：B

知识点：最小公倍数整除性

难度评级：1070

小提示：

设成人和儿童的相同人数为

P

。

Let the equal number of adults and children be

P

大提示：

长凳单元数为

\frac{P}{7}+\frac{P}{11}

。

The number of bench sections is

\frac{P}{7}+\frac{P}{11}

视频讲解：

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文字解答：

设成人和儿童各有

P

人。成人需要

\frac{P}{7}

个长凳单元，儿童需要

\frac{P}{11}

个长凳单元，因此

N=P\left(\dfrac17+\dfrac1{11}\right)=\dfrac{18P}{77}

。

使其为整数的最小值是

P=77

，此时

N=18

。正确答案是 B。

If the equal number of adults and children is

P

, then the adults use

\frac{P}{7}

bench sections and the children use

\frac{P}{11}

bench sections. Thus

N=P\left(\dfrac17+\dfrac1{11}\right)=\dfrac{18P}{77}

.

The least positive integer occurs when

P=77

, giving

N=18

. Thus, B is the correct answer.