2006 AMC 8 Problems/Problem 23

Problem

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 ¯ve 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? (A) 0 (B) 1 (C) 2 (D) 3 (E) 5

Solution

(A) The counting numbers that leave a remainder of 4 when divided by 6 are 4; 10; 16; 22; 28; 34; : : :. The counting numbers that leave a remainder of 3 when divided by 5 are 3; 8; 13; 18; 23; 28; 33; : : :. So 28 is the smallest possible number of coins that meets both conditions. Because 4 £ 7 = 28, there are no coins left when they are divided among seven people. OR If there were two more coins in the box, the number of coins would be divisible by both 6 and 5. The smallest number that is divisible by 6 and 5 is 30, so the smallest possible number of coins in the box is 28.