What is the smallest two-digit positive integer $k$ for which the product $45 k$ is a perfect square?
We note that $45=3^{2} \cdot 5$ and so $45 k=3^{2} \cdot 5 \cdot k$.

For $45 k$ to be a perfect square, each prime factor has to occur an even number of times. Therefore, $k$ must be divisible by 5 .

We try the smallest two-digit possibilities for $k$ that are divisible by 5 , namely $k=10,15,20, \ldots$

If $k=10$, then $45 k=450$, which is not a perfect square.

If $k=15$, then $45 k=675$, which is not a perfect square.

If $k=20$, then $45 k=900=30^{2}$, which is a perfect square.

Therefore, the smallest two-digit positive integer $k$ for which $45 k$ is a perfect square is $k=20$.