Obviously?
Find the values of $n$ for which $1^n + 8^n - 3^n - 6^n$ is divisible by 6.
Consider separately when $N = 1^n + 8^n - 3^n - 6^n$ is divisible by $2$ and when it is divisible by $3$.
Murat from Turkey sent the following solution.
If $g(n)=(1+8^n-3^n)$ is divisible by 6, then $(1+8^n-3^n-6^n)$ is also divisble by 6. It can be verified that
$$g(n)-73g(n-2)+576g(n-4)=504$$
for all positive integers $n> 3$. Since $g(1)$ and $g(3)$ are divisible by 6, it follows that $g(5)$ is also. By induction, it can be shown that for all odd $n$, $g(n)$ is divisible by 6. Since $g(2)$ and $g(4)$ are not divisible by 6, this is not the case for even $n$.
An alternative approach is to use the facts that powers of odd numbers are always odd; powers of even numbers are always even; also the difference of two odd numbers is even. Hence $N=1^n+8^n-3^n-6^n$ is even (odd + even - odd - even).
It remains to decide whether or not $N$ is divisible by 3.
$N \equiv 1 + (-1)^n - 0 - 0$ (mod 3)
This shows that $N \equiv 0$ (mod 3) if $n$ is odd and hence $N$ will be divisible by 6 for all odd values of $n$. However, $N \equiv 2$ (mod 3) if $n$ is even and so $N$ cannot be divisible by 6 for even values of $n$.