Find all $n \in \Bbb Z$ such that $n^2 + n + 1$ divide $n^3-22$

Question

I need help with this problem:

Find all $n \in \Bbb Z$ such that $n^2 + n + 1$ divide $n^3-22$.

I've got to a point where I know that $n^2 + n + 1 | -21$. So it should be among {${-21, -7, -3, -1, 1, 3, 7, 21}$}.

How could I continue? Thanks

Answer 1

Check for all eight of these numbers whether they satisfy the given property. The ones that do are your solution.