<s>[INST] How to prove $n^5 - n$ is divisible by $30$ without reduction How can I prove that prove $n^5 - n$ is divisible by $30$?
I took $n^5 - n$ and got $n(n-1)(n+1)(n^2+1)$
Now, $n(n-1)(n+1)$ is divisible by $6$.
Next I need to show that $n(n-1)(n+1)(n^2+1)$ is divisible by $5$.
My guess is using Fermat's little theorem but I don't know how.
 [/INST] $$n^2+1\equiv n^2+5n+6\mod 5$$ and $$n^2+5n+6=(n+2)(n+3).$$
Then
$$(n-1)n(n+1)(n+2)(n+3)$$ is divisible by $5$.
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