2016 AMC 12B Problem 24

There are exactly 77,000 ordered quadruplets (a, b, c, d) such that \gcd(a, b, c, d) = 77 and \text{lcm}(a, b, c, d) = n. What is the smallest possible value for n?

\textbf{(A)}\ 13,860\qquad\textbf{(B)}\ 20,790\qquad\textbf{(C)}\ 21,560 \qquad\textbf{(D)}\ 27,720 \qquad\textbf{(E)}\ 41,580

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