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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?

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This problem is taken from the UKMT Mathematical Challenges.