2012 AMC 10B Problem 4

Question

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?

1

2

3

4

5

Add the two remainders modulo

6

4+3

leaves remainder

1

when divided by

6

As we know that when Ringo’s marbles are divided by

6,

we have a remainder of

4,

we conclude that he has

6x+4

marbles for some

x.

Using the same logic, we can also conclude that Paul has

6y+3

marbles for some

y.

Therefore, the total number of marbles is

(6x+4)+(6y+3)

= 6(x+y+1)+1

which, when divided by

6,

only leaves

1

left over.

Thus, the correct answer is A .

Problem 4 in other years