2010 AMC 12B Problem 3

Attempt Problem 3 of the 2010 AMC 12B below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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?

$1$

$2$

$3$

$4$

$5$

Answer: E

Concepts:greatest common divisor factor counting

Difficulty rating: 1010

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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