Problem:
A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The pirate to take a share takes of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the pirate receive?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
For , the number of coins remaining in the chest before the pirate takes a share is times the number remaining afterward. Thus if there are coins left for the pirate to take, the number of coins originally in the chest is
The smallest value of for which this is a positive integer is . In this case there are
coins left for the pirate to take, and note that this amount is an integer for each . Hence the pirate receives coins.
The problems on this page are the property of the MAA's American Mathematics Competitions