2012 AMC 10B Problem 4

P1-10~1 minPrint

Modular Arithmetic Set up the equation/formula and compute; no special trick needed

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 left over?

Show solution without answering

Key insight

Put the two leftover piles together: 4 + 3 = 7 marbles. Seven marbles fill one more bag of 6, leaving 1.

Solution

Ringo's marbles make some full bags of $6$ , plus $4$ extra marbles.
Paul's marbles make some full bags of $6$ , plus $3$ extra marbles.
When they pool everything, the full bags are still full, so they don't matter. Only the extras need new bags: $4 + 3 = 7$ extra marbles.
Seven marbles fill one more bag of $6$ , with $7 - 6 = 1$ marble left over.

Try it with real numbers: Ringo has $10$ marbles ( $6 + 4$ ) and Paul has $9$ ( $6 + 3$ ). Together that is $19 = 3 \times 6 + 1$ . One marble is left over.

The answer is $\boxed{\textbf{(A)}\ 1}$ .

Why this works

The leftover of a sum depends only on the leftovers of the parts. Add the leftovers, and if the result is $6$ or more, take out another full bag. Mathematicians write this as $4 + 3 = 7 \equiv 1 \pmod 6$ , which just means " $7$ leaves the same leftover as $1$ when you divide by $6$ ."

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Ringo's marbles fill some whole bags plus $4$ extra; Paul's fill some whole bags plus $3$ extra. When they pool everything, the full bags stay full, and the $4 + 3 = 7$ leftover marbles are combined.

Seven marbles fill one more bag of $6$ with $1$ marble remaining.

In congruence language: $R \equiv 4$ and $P \equiv 3 \pmod 6$ , so $R + P \equiv 7 \equiv 1 \pmod 6$ .

The answer is $\boxed{\textbf{(A)}\ 1}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 12: Introduction to Modular Arithmetic
Introduction to Number Theory, Chapter 10: Units Digits
Introduction to Number Theory, Chapter 14: Linear Congruences (§14.4)