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2009 AMC 12A Problem 4

Problem 4 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

If the four coins include a penny, the total is not a multiple of 5,5, so it cannot equal any of the five listed multiples of 5.5. If there is no penny, every coin is worth at least 55 cents, so the total is at least 2020 cents. Either way, 1515 is impossible. The other amounts are attainable: 25=10+5+5+5,25 = 10 + 5 + 5 + 5, 35=10+10+10+5,35 = 10 + 10 + 10 + 5, 45=25+10+5+5,45 = 25 + 10 + 5 + 5, and 55=25+10+10+10.55 = 25 + 10 + 10 + 10. Thus, the correct answer is A.

More practice

Concepts: money · multiple · casework

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