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2000 AMC 10 Problem 17

Problem 17 of 25IntermediateNumber TheoryArithmeticProblem-Solving Techniques

Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?

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Solution

Trading a quarter for five nickels or a nickel for five pennies does not change the total value. Only trading a penny for five quarters changes it, adding 5⋅25−1=1245 \cdot 25 - 1 = 124 cents. Starting from 11 cent, Boris always has 1+124n1 + 124n cents for some nonnegative integer n.n. Only $7.45\$7.45 has this form, since 745=1+124⋅6.745 = 1 + 124 \cdot 6. Thus, the correct answer is D.
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