1957 AHSME Problem 32

The largest of the following integers which divides each of the numbers of the sequence 1^5 - 1,\, 2^5 - 2,\, 3^5 - 3,\, \cdots, n^5 - n, \cdots is:

\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 60 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 120\qquad \textbf{(E)}\ 30

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