2003 AMC 12B Problem 12

What is the largest integer that is a divisor of (n+1)(n+3)(n+5)(n+7)(n+9) for all positive even integers n?

\text {(A) } 3 \qquad \text {(B) } 5 \qquad \text {(C) } 11 \qquad \text {(D) } 15 \qquad \text {(E) } 165

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