Modular congruence, divisibility

Question

Show that, if $7$ $\nmid$ $n$, $7 \nmid n-1$ and $7 \nmid n^{3}+1$ then $7 \mid n^{2}+n+1$.

I tried for modular congruence, but i could not.

Answer 1

Using Fermat's little theorem, $7$

divides $$n^7-n=n(n-1)(n^2+n+1)(n^3+1)$$ for any integer $n$