Question

Ms. Demeanor 班上 $30$ 名学生中的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（单位：美分）大于每名学生买的铅笔数量，所有铅笔总费用为 $\$17.71$ 。每支铅笔的价格是多少美分？

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：质因数分解整除性极限情形界定

难度评级：1370

解答：

总费用为 $1771$ 美分，且 $1771 = 7 \cdot 11 \cdot 23$ 。写成（学生数）（每人铅笔数）（每支铅笔价格） $= 1771$ ，学生数必须是该数的一个因数，并且是 $30$ 人中的多数，即超过 $15$ 。唯一这样的因数是 $23$ 。

因此（铅笔数）（单价） $= 77 = 7 \cdot 11$ ，且单价 $\gt$ 铅笔数 $\gt 1$ ，所以每人买 $7$ 支铅笔，每支 $11$ 美分。

因此，正确答案是 B。

Total cents is $1771 = 7 \cdot 11 \cdot 23.$ Writing (students)(pencils each)(cost per pencil) $= 1771,$ the number of students is a divisor of $1771$ that is a majority of $30,$ hence more than $15.$ The only such divisor is $23.$

Then (pencils)(cost) $= 77 = 7 \cdot 11$ with cost $\gt$ pencils $\gt 1,$ forcing $7$ pencils at $11$ cents each.

Thus, the correct answer is B.

2011 AMC 12A 第 7 题

答案：B

难度评级：1370

解答：

其他年份的第 7 题