AMC 12 Mock Exam 4 Problem 6

Write $a=6x$, $b=6y$ with $\gcd(x,y)=1$. Then $36xy=2592=2^5\cdot3^4$, so $xy=2^3\cdot3^2=72$. Since $x$ and $y$ are coprime, the factor $2^3$ goes entirely to $x$ or entirely to $y$, and likewise for $3^2$. That gives $2\cdot2=\boxed{4}$ ordered pairs: $(x,y)=(1,72),(8,9),(9,8),(72,1)$.