Question

A box contains gold coins. If the coins are equally divided among six people, four coins are left over. If the coins are equally divided among five people, three coins are left over. If the box holds the smallest number of coins that meets these two conditions, how many coins are left when equally divided among seven people?

$0$

$1$

$2$

$3$

$5$

Answer: A

Concepts:modular arithmetic systematic listing

Difficulty rating: 1450

Solution:

The positive integers that leave a remainder of $4$ when divided by $6$ are $4, 10, 16, 22, 28, 34, 40, \cdots.$ The positive integers that leave a remainder of $3$ when divided by $5$ are $3, 8, 13, 18, 23, 28, 33, \cdots.$

From this, we can see that the smallest number of coins that work is $28.$ This leaves a remainder of $0$ when divided by $7.$

Thus, A is the correct answer.

2006 AMC 8 Problem 23

Answer: A

Difficulty rating: 1450

Solution:

Problem 23 in Other Years