Prove that $0$ divides $a$ if and only if $a = 0$

Question

Let $a \in \mathbb{Z}$. Prove that $0\mid a$ if and only if $a = 0$.

I am a first year maths student and I'm looking for a well-constructed proof for this statement that uses the fact that..

Let $a, b \in \mathbb{Z}$. $a\mid b$ if there exists some $k \in \mathbb{Z}$ that satisfies $b = ka$.

Answer 1

CASE 1

Suppose that 0 divides a. By the definition of ‘divides’, there is an integer q such that a = 0q, so that a = 0q = 0, as desired.

CASE 2

Suppose that a = 0. Then we have that a = 0(1), so that 0 divides a, as desired

UNDERSTOOD