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Integer and Integer

Age 14 to 16 Short Challenge Level:

For how many values of $n$ are both $n$ and $\frac{n+3}{n-1}$ integers?

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.