Euclid's Lemma

Theorem
Let $$a,b,c \in \mathbb{Z}$$.

If $$a|bc$$, where $$a$$ and $$b$$ are relatively prime, then $$a|c$$.

Proof
$$a \perp b$$ from the definition of relatively prime.

That is, $$\gcd \left\{{a, b}\right\} = 1$$.

From Integer Combination of Coprime Integers, we may write $$ax+by=1$$ for some $$x,y \in \mathbb{Z}$$.

Upon multiplication by $$c$$, we see that $$c=c(ax+by)=cax+cby$$.

Since $$a|ac$$ and $$a|bc$$, it is clear that $$a|(cax+cby)$$.

However, $$cax+cby=c(ax+by)=c \cdot 1=c$$. Therefore, $$a|c$$.