2006 AMC 12A Problems/Problem 9
Problem
Oscar buys pencils and erasers for . A pencil costs more than an eraser, and both items cost a whole number of cents. What is the total cost, in cents, of one pencil and one eraser?
Solution
Let the price of a pencil be and an eraser . Then with . Since and are positive integers, we must have and .
Considering the equation modulo 3 (that is, comparing the remainders when both sides are divided by 3) we have so leaves a remainder of 1 on division by 3.
Since , possible values for are 4, 7, 10 ....
Since 13 pencils cost less than 100 cents, . which is too high, so must be 4 or 7.
If then and so giving . This contradicts the pencil being more expensive. The only remaining value for is 7; then the 13 pencils cost cents and so the 3 erasers together cost 9 cents and each eraser costs cents.
Thus one pencil plus one eraser cost cents, which is answer choice .