2011 AMC 12A Problem 7

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 bounding to limit cases

Difficulty rating: 1370

Small hint:

Factor

1771 = 7 \cdot 11 \cdot 23

and let (students)(pencils)(cost)

= 1771

Big hint:

A majority means more than

15

students, so the number of students is

23

Solution:

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.