AMC 12 2016b Problem 24

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

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