2010 AMC 12B Problem 3

Below is the professionally curated solution for Problem 3 of the 2010 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2010 AMC 12B solutions, or check the answer key.

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Concepts:greatest common divisor factor counting

Difficulty rating: 1010

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:

The price $x$ must divide both totals, so $x$ is a common divisor of $48$ and $64.$

Since $\gcd(48,64)=16,$ the common divisors are $1, 2, 4, 8,$ and $16.$ There are $5$ possible values.

Thus, the correct answer is E.

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