Congruent to Zero iff Modulo is Divisor

Theorem
Let $a, z \in \R$.

Then $a$ is congruent to $0$ modulo $z$ $a$ is an integer multiple of $z$.


 * $\exists k \in \Z: k z = a \iff a \equiv 0 \pmod z$

If $z \in \Z$, then further:
 * $z \divides a \iff a \equiv 0 \pmod z$

Proof
Thus by definition of congruence, $a \equiv 0 \pmod z$ and the result is proved.

If $z$ is an integer, then by definition of divisor:
 * $z \divides a \iff \exists k \in \Z: a = k z$

Hence the result for integer $z$.