AMC 12 2016a Problem 18

For some positive integer $n,$ the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3.$ How many positive integer divisors does the number $81n^4$ have?

\textbf{(A)}\ 110 \textbf{(B)}\ 191 \textbf{(C)}\ 261 \textbf{(D)}\ 325 \textbf{(E)}\ 425