Question

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?

$3.63

$5.13

$6.30

$7.45

$9.07

Answer: D

Concepts:invariant modular arithmetic money

Difficulty rating: 1750

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 $5 \cdot 25 - 1 = 124$ cents.

Starting from $1$ cent, Boris always has $1 + 124n$ cents for some nonnegative integer $n.$

Only $\$7.45$ has this form, since $745 = 1 + 124 \cdot 6.$

Thus, the correct answer is D.

2000 AMC 10 Problem 17

Answer: D

Difficulty rating: 1750

Solution:

Problem 17 in Other Years