AMC10 2011 A

AMC10 2011 A · Q10

AMC10 2011 A · Q10. It mainly tests Money / coins, Primes & prime factorization.

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?

Demeanor女士班上有30名学生，大多数学生在学校书店买了铅笔。这些学生每人都买了相同数量的铅笔，且这个数量大于1。铅笔的单价（以美分计）大于每个学生买的铅笔数量，所有铅笔的总价是\$17.71。铅笔的单价（以美分计）是多少？

(A) 7 7

(B) 11 11

(D) 23 23

(E) 77 77

Answer

Correct choice: (B)

正确答案：（B）

Solution

Let $C$ be the cost of a pencil in cents, $N$ be the number of pencils each student bought, and $S$ be the number of students who bought pencils. Then $C \cdot N \cdot S = 1771 = 7 \cdot 11 \cdot 23$, and $C > N > 1$. Because a majority of the students bought pencils, $30 \ge S > \frac{30}{2} = 15$. Therefore $S = 23$, $N = 7$, and $C = 11$.

设$C$为铅笔单价（美分），$N$为每个学生买的铅笔数，$S$为买铅笔的学生数。那么$C \cdot N \cdot S = 1771 = 7 \cdot 11 \cdot 23$，且$C > N > 1$。因为大多数学生买了铅笔，$30 \ge S > \frac{30}{2} = 15$。因此$S = 23$，$N = 7$，$C = 11$。

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