I need to find for which values of $a\in\mathbb{N}$ the following happens:

$$a-2\ |\ a^3+4$$

that is, for which values of $k\in\mathbb{Z}$ the following holds:

$$a^3+4 = k(a-2)$$

I could not find anything useful from that.

Usually for a proof involving division, for example, proving $8|3^{2n}+7$ I'd suppose $3^{2k}+7 = 8k$ and prove by induction that it also holds for $k+1$, but for this case is diferent.