2005 AMC 10B Problem 22

For how many positive integers n less than or equal to 24 is n! evenly divisible by 1 + 2 + \ldots + n?

\mathrm{(A)} 8 \qquad \mathrm{(B)} 12 \qquad \mathrm{(C)} 16 \qquad \mathrm{(D)} 17 \qquad \mathrm{(E)} 21

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