For which values of $a$ does $a-2 \mid a^3+4$ [closed]
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.
Solution 1:
Hint : $$a^3+4=(a-2)(a^2+2a+4)+12$$
Solution 2:
Hint: $a-2\mid a^3-8$. When can $a-2\mid a^3+4$?