How would I prove for all a that a divides zero

Question

I'm trying to prove for all a such that a divides zero. I can explain verbally why it works but I can't seem to be able to write it down in "proof" form. Could someone help me out?

Answer 1

$a\ |\ b$ iff $\exists\ k \in \mathbb{Z}$ such that $b = ka$.

Let $a$ be arbitrary and $b = 0$. Then, $a\ |\ 0$ since $0\cdot a = 0\ \forall\ a$.

Answer 2

$0\cdot a=0$ seems to do the trick.

Or if you have just that $0$ is the additive identity and the distributive law, with $1$ the multiplicative identity (in a ring or a field) you can say $$a \cdot 0 = a\cdot (1-1)=a-a=0$$