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?

$7$

$11$

$17$

$23$

$77$

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.

2011 AMC 10A Problem 10

Solution:

Problem 10 in Other Years