### Instruction: How many factors of $2^5\cdot3^6$ are perfect squares?### Response: A factor of $2^5\cdot3^6$ is of the form $2^a\cdot3^b$, where $0\leq a\leq5$ and $0\leq b\leq6$.
For it to be a perfect square, both $a$ and $b$ must be even.
So I need to count how many even values are there for $a$ and $b$ in their respective ranges.
For $a$, the even values are $0,2,4$, so there are $3$ possibilities.
For $b$, the even values are $0,2,4,6$, so there are $4$ possibilities.