2016 AMC 10A Problem 25

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

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