2011 AMC 10A 第 10 题

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

小提示：

分解

1771

。

Factor

1771

大提示：

买铅笔的人数是

30

人中的多数，所以使用大于

15

的因数。

The number of students is a majority of

30

, so use the factor greater than

15

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\text{。}

还必须满足

30 \geq s \gt 15, p \gt 1, c \gt p\text{。}

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

小提示：

大提示：

解答：

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