2010 AMC 10B Problem 8

Below is the professionally curated solution for Problem 8 of the 2010 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 10B solutions, or check the answer key.

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Concepts:greatest common divisor factor counting

Difficulty rating: 1020

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?

$1$

$2$

$3$

$4$

$5$

Solution:

Note that $x$ must divide both $48$ and $64,$ which means that means 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.

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