# Product of Coprime Factors

## Theorem

Let $a, b, c \in \Z$ such that $a$ and $b$ are coprime.

Let both $a$ and $b$ be divisors of $c$.

Then $a b$ is also a divisor of $c$.

That is:

$a \perp b \land a \divides c \land b \divides c \implies a b \divides c$

## Proof

We have:

$a \divides c \implies \exists r \in \Z: c = a r$
$b \divides c \implies \exists s \in \Z: c = b s$

So:

 $\displaystyle a$ $\perp$ $\displaystyle b$ $\displaystyle \leadsto \ \$ $\displaystyle \exists m, n \in \Z: m a + n b$ $=$ $\displaystyle 1$ Integer Combination of Coprime Integers $\displaystyle \leadsto \ \$ $\displaystyle c m a + c n b$ $=$ $\displaystyle c$ $\displaystyle \leadsto \ \$ $\displaystyle b s m a + a r n b$ $=$ $\displaystyle c$ $\displaystyle \leadsto \ \$ $\displaystyle a b \paren {s m + r n}$ $=$ $\displaystyle c$ $\displaystyle \leadsto \ \$ $\displaystyle a b$ $\divides$ $\displaystyle c$

$\blacksquare$