2011 AMC 10A Problem 10

Question

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

Answer: B

Concepts:prime factorization divisibility

Difficulty rating: 1420

Small hint:

Factor

1771

Big hint:

The number of students is a majority of

30

, so use the factor greater than

15

Solution:

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.