Prove that for odd $n > 1$ , $3^{n} + 1$ is not divisible by $n$.
There's a hint but I can't find any use for that.
hint: If $a$ and $b$ are coprime with $m$ and $a^{x} \equiv b^{x}$ (mod $m$) and $a^{y} \equiv b^{y}$ (mod $m$) then $a^{\gcd(x,y)} \equiv b^{\gcd(x,y)}$ (mod $m$).
I don't know if you can find any use for this but this is my thoughts : $3$ and $n-1$ are coprime with $n$ and $3^{n} \equiv (n-1)^{n} \equiv -1$ (mod $n$).