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AMC10每日一题（2000年真题#17）

2018-04-25
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2000 AMC 10 竞赛试题/第17题

Problem

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?

$\mathrm{(A)}$ $$3.63$

$\mathrm{(B)}$ $$5.13$

$\mathrm{(C)}$ $$6.30$

$\mathrm{(D)}$ $$7.45$

$\mathrm{(E)}$ $$9.07$

Solution

Consider what happens each time he puts a coin in. If he puts in a quarter, he gets five nickels back, so the amount of money he has doesn't change. Similarly, if he puts a nickel in the machine, he gets five pennies back and the money value doesn't change. However, if he puts a penny in, he gets five quarters back, increasing the amount of money he has by $124$ cents.

This implies that the only possible values, in cents, he can have are the ones one more than a multiple of $124$ . Of the choices given, the only one is $\boxed{\text{D}}$