# 2000 AMC 10 Problems/Problem 17

## 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?

$\textbf{(A)} 3.63 \qquad \textbf{(B)} 5.13 \qquad \textbf{(C)}6.30 \qquad \textbf{(D)} 7.45 \qquad \textbf{(E)} 9.07$

## 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 $\boxed{\text{D}}$

~savannahsolver

-gnv12