How to show $n^5 + 29 n$ is divisible by $30$ Show that $n^5 + 29 n$ is divisible by $30$.
Attempt:
$n^4 ≡ 1 \pmod 5$ By Fermat Little Theorem

Modulo $6$, the expression is $n^5 + (-1)n = n^5 -n$.
Modulo $5$, $n^5 \equiv n$ by Fermat's Little Theorem, so $n^5 - n \equiv 0 \pmod 5$.
Since $5$ and $6$ are coprime $n^5 + 29n \equiv 0 \pmod {30}$.
