Square LCM

Using the hcf and lcf of the numerators, can you deduce which of these fractions are square numbers?
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Problem



The highest common factor of two positive integers $m$ and $n$ is $12$, and their lowest common multiple is a square number.

How many of the five numbers $\frac{n}{3}$, $\frac{m}{3}$, $\frac{n}{4}$, $\frac{m}{4}$ and $mn$ are square numbers?

If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.