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 $\boxed{\textbf{(A)}\ 1}$ marble left over.

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2012 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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2012 AMC 10B Problems/Problem 4

Contents

Problem 4

Solution 1

Solution 2 (modulo)

See Also