Show that $7 |(n^6 + 6)$ if $7 ∤ n$, $∀ n ∈ ℤ$ 
Show that $7 |(n^6 + 6)$ if $7 ∤ n$, $∀ n ∈ ℤ$

I need to prove this by the little Fermat's theorem.
My attempt
$n^6 \equiv -6 \pmod 7$
To show $7  ∤ n$ I need to show that $N$ is not congruent to $0$ mod $7$.
as $-6 \equiv 1\pmod 7$
$n^6 \equiv 1\pmod7$
But now, How can I show $N$ is not congruent to $0$ mod $7$ ?

Because $n^6+6=n^6-1+7$ and  $n^6-1$ divisible by $7$.
$$n^6-1=(n-1)(n+1)(n^2-n+1)(n^2+n+1)$$
and now easy to check $n\equiv\pm1,\pm2\pm3(\mod7)$.
