CMOQR 2013 Problem 3

A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$ , $\text{lcm}(x, z) = 900$ , and $\text{lcm}(y, z) = n$ , where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.