2005 AMC 12B Problem 21

A positive integer n has 60 divisors and 7n has 80 divisors. What is the greatest integer k such that 7^k divides n?

\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}

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