Question

Show that $n^5-n$ is divisible by $30$ for all values of $n$, and by $240$ if $n$ is odd.

Answer 1

It is known that $n^p-n$ is a multiple of $p$

Therefore, $n^5-n$ is divisible by $5$.

$\Rightarrow n^5-n=n(n^4-1)=n(n-1)(n+1)(n^2+1)$

And this expression is divisible by $31$ or $6$

Thus $n^5-n$ is divisible by $5.6=30$

Again if $n$ is odd the above expression is divisible by $240$.