Question

Demeanor 老师班上 $30$ 名学生中的大多数人在学校书店买了铅笔。这些学生每人买了相同数量的铅笔，且这个数量大于 $1$ 。每支铅笔的价格（美分）大于每人购买的铅笔数。所有铅笔的总价为 $\$17.71$ 。每支铅笔多少美分？

A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1.$ The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $\$17.71.$ What was the cost of a pencil in cents?

$7$

$11$

$17$

$23$

$77$

答案：B

知识点：质因数分解整除性

难度评级：1420

解答：

设每人买 $p$ 支铅笔，买铅笔的学生有 $s$ 人，每支铅笔 $c$ 美分。

则 $psc = 1771 = 7 \cdot 11 \cdot 23.$

还必须满足 $30 \geq s \gt 15, p \gt 1, c \gt p.$

由质因数分解，唯一满足学生人数条件的是 $s = 23$ 。

剩下的因数是七和十一；结合价格大于购买数量的条件，得到 $p = 7$ 、 $c = 11$ 。

所以正确答案是 B。

Let $p$ be the number of pencils that each student bought, $s$ be the number of students that bought pencils, and $c$ be the cost of a pencil.

We have that $psc = 1771 = 7 \cdot 11 \cdot 23.$

We also have the following restrictions: $30 \geq s \gt 15, p \gt 1, c \gt p.$

From the above prime factorization, we have that $s = 23$ is the only value that satisfies the conditions.

Finally, we get that $p = 7$ and $c = 11$ are the only remaining values that satisfy the other conditions.

Thus, B is the correct answer.

2011 AMC 10A 第 10 题

答案：B

难度评级：1420

解答：

其他年份的第 10 题