Euclid's Lemma

Theorem
Let $$a, b, c \in \Z$$.

If $$a \backslash b c$$, where $$a$$ and $$b$$ are relatively prime, then $$a \backslash c$$.

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 $$a x + b y = 1$$ for some $$x, y \in \Z$$.

Upon multiplication by $$c$$, we see that $$c = c \left({a x + b y}\right) = c a x + c b y$$.

Since $$a \backslash a c$$ and $$a \backslash b c$$, it is clear that $$a \backslash \left({c a x + c b y}\right)$$.

However, $$c a x + c b y = c \left({a x + b y}\right) = c \cdot 1 = c$$.

Therefore, $$a \backslash c$$.