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2010 AMC 12B Problem 3

Problem 3 of 25EasierNumber Theory

A ticket to a school play costs xx dollars, where xx is a whole number. A group of 99th graders buys tickets costing a total of $48,\$48, and a group of 1010th graders buys tickets costing a total of $64.\$64. How many values for xx are possible?

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Solution

The price xx must divide both totals, so xx is a common divisor of 4848 and 64.64. Since gcd⁡(48,64)=16,\gcd(48,64)=16, the common divisors are 1,2,4,8,1, 2, 4, 8, and 16.16. There are 55 possible values. Thus, the correct answer is E.

More practice

Concepts: greatest common divisor · factor counting

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.