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Square LCM
Age 14 to 16
ShortChallenge Level 
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.
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.
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