Congruent to Zero iff Modulo is Divisor

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

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


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

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

Proof
$$ $$

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

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

Hence the result for integral $$z$$.