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2009 AMC 10A Problem 2

Problem 2 of 25EasierAlgebraNumber Theory

Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes, and quarters. Which of the following could not be the total value of the four coins, in cents?

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Solution

To get a multiple of 55 cents, the number of pennies must be a multiple of 5.5. With only four coins, that means using no pennies, but then the four coins are each worth at least 55 cents, for a total of at least 2020 cents. So 1515 cents cannot be made. The others can: 25=10+3(5),25 = 10 + 3(5), 35=3(10)+5,35 = 3(10) + 5, 45=25+10+2(5),45 = 25 + 10 + 2(5), and 55=25+3(10).55 = 25 + 3(10). Thus, the correct answer is A.

More practice

Concepts: money · divisibility · bounding to limit cases

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