What is the largest integer $n$ with the properties that $200<n<250$ and that $12 n$ is a perfect square?
Suppose $12 n=k^{2}$ for some integer $k$. Then $k^{2}$ is even and so $k$ must be even, which means $\frac{k}{2}$ is an integer.

Dividing both sides of $12 n=k^{2}$ by 4 gives $3 n=\left(\frac{k}{2}\right)^{2}$, and since $\frac{k}{2}$ is an integer, this means $3 n$ is a perfect square.

We are given that $200<n<250$, which implies $600<3 n<750$.

The perfect squares between 600 and 750 are 625,676 , and 729 , among which 729 is the only multiple of 3 , so $3 n=729$ or $n=243$.

This value of $n$ satisfies $200<n<250$, and $12 n=2916=54^{2}$.

We have shown that $n=243$ is the only $n$ that satisfies the conditions, so it must be the largest.