2010 AMC 10B Problem 8

Question

A ticket to a school play costs

x

dollars, where

x

is a whole number. A group of

9

th graders buys tickets costing a total of

\$48,

and a group of

10

th graders buys tickets costing a total of

\$64.

How many values for

x

are possible?

1

2

3

4

5

The ticket price must divide both totals

Count the positive divisors of

\gcd(48,64)

Note that

x

must divide both

48

and

64,

which means that it must divide their greatest common divisor.

The greatest common divisor of

48

and

64

is

16,

which means

x

divides

16.

16

has

5

factors, namely all the powers of

2

up to

16.

Thus, E is the correct answer.

Problem 8 in other years