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

You can prove this by induction. Case $n = 1$ is obviously true.
Hint: Use the binomial theorem in the inductive step. Also, a number is divisible by $30$ if and only if it is divisible by its prime factors $2$, $3$, and $5$.
