# Difference between revisions of "2012 AMC 10B Problems/Problem 4"

## Problem 4

When Ringo places his marbles into bags with 6 marbles per bag, he has 4 marbles left over. When Paul does the same with his marbles, he has 3 marbles left over. Ringo and Paul pool their marbles and place them into as many bags as possible, with 6 marbles per bag. How many marbles will be leftover?

$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

## Solution 1

In total, there were $3+4=7$ marbles left from both Ringo and Paul.We know that $7 \equiv 1 \pmod{6}$. This means that there would be $1$ marble leftover, or $\boxed{A}$.

## Solution 2 (modulo)

Let $r$ be the number of marbles Ringo has and let $p$ be the number of marbles Paul has. we have the following equations: $$r \equiv 4 \mod{6}$$ $$p \equiv 3 \mod{6}$$ Adding these equations we get: $$p + r \equiv 7 \mod{6}$$ We know that $7 \equiv 1 \mod{6}$ so therefore: $$p + r \equiv 7 \equiv 1 \mod{6} \implies p + r \equiv 1 \mod{6}$$ Thus when Ringo and Paul pool their marbles, they will have $\boxed{\textbf{(A)}\ 1}$ marbles left over.

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 2012 AMC 10B (Problems • Answer Key • Resources) Preceded byProblem 3 Followed byProblem 5 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions

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