Converse of Euclid's Lemma does not Hold

Lemma
Let $n \in Z_{>0}$ be a positive composite number

Let $a b \equiv 0 \pmod n$.

Then it is not necessarily the case that either $a \equiv 0 \pmod n$ or $b \equiv 0 \pmod n$.

Proof
Let $n = 6$.

We have that:
 * $6 \equiv 0 \pmod 6$

but:
 * $2 \equiv 2 \pmod 6$
 * $3 \equiv 3 \pmod 6$

Hence the result by Proof by Counterexample.