smc

Question

A group of 12 pirates agree to divide a treasure chest of gold coins among themselves as follows. The k^th pirate to take a share takes k12 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 12^th pirate receive?

Accepted Answer

The first pirate takes 112 of the x coins, leaving 1112 x. The second pirate takes 212 of the remaining coins, leaving 1012 1112*x. Note that 12^11 = (2^2 3)^11 = 2^22 3^11 11! = 11 10 9 8 7 6 5 4 3 2 All the 2s and 3s cancel out of 11!, leaving 11 5 7 5 = 1925 in the numerator. We know there were just enough coins to cancel out the denominator in the fraction. So, at minimum, x is the denominator, leaving 1925 coins for the twelfth pirate.