AMC 12 2016a Problem 22

How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600$ and $\text{lcm}(y,z)=900$?

\textbf{(A)}\ 15 \textbf{(B)}\ 16 \textbf{(C)}\ 24 \textbf{(D)}\ 27 \textbf{(E)}\ 64