<s>[INST] 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
 [/INST] You want to show that $n^5+29n$ is divisible by $2$, $3$ and $5$.


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*$n^5+29n\equiv n^5+n\equiv n^2n^2n+n\equiv nnn+n\equiv nn+n\equiv n+n\equiv0\pmod{2}$

*$n^5+29n\equiv n^5-n\equiv n^3n^2-n\equiv nn^2-n\equiv n-n\equiv0\pmod{3}$

*$n^5+29n\equiv n^5-n\equiv n-n\equiv0\pmod{5}$
All use Fermat's little theorem: $n^p\equiv n\pmod{p}$ for $p$ prime.
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