2013 AMC 12A Problem 17

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

\textbf{(A)} \ 720 \qquad \textbf{(B)} \ 1296 \qquad \textbf{(C)} \ 1728 \qquad \textbf{(D)} \ 1925 \qquad \textbf{(E)} \ 3850

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.