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 $n$?
$\text {(A) } 3 \qquad \text {(B) } 5 \qquad \text {(C) } 11 \qquad \text {(D) } 15 \qquad \text {(E) } 165$
For all consecutive odd integers, one of every five is a multiple of 5 and one of every three is a multiple of 3. The answer is $3 \cdot 5 = 15$, so  ${\boxed{\textbf{(D)15}}}$ is the correct answer.