Question:
How many factors of $2^5\cdot3^6$ are perfect squares?

Answer:
All factors of $2^5\cdot 3^6$ that are perfect squares must be in the form $(2^m\cdot 3^n)^2=2^{2m}\cdot 3^{2n}$, where $0\le2m\le5$ and $0\le2n\le6$ for integers $m$ and $n$. Thus, $0\le m\le2$ and $0\le n\le3$, for a total of $3\cdot4=\boxed{12}$ factors that are perfect squares.