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?

$1$

$2$

$3$

$4$

$5$

Answer: A

Concepts:modular arithmetic

Difficulty rating: 770

Solution:

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 .

2012 AMC 10B Problem 4

Answer: A

Difficulty rating: 770

Solution:

Problem 4 in Other Years