Product of Coprime Factors

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Theorem

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

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

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

That is:

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

We have:

So:

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle a$	$\perp$	$\displaystyle b$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle \exists m, n \in \Z: m a + n b$	$=$	$\displaystyle 1$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Integer Combination of Coprime Integers
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle c m a + c n b$	$=$	$\displaystyle c$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle b s m a + a r n b$	$=$	$\displaystyle c$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle a b \left({s m + r n}\right)$	$=$	$\displaystyle c$	$\displaystyle $	$\displaystyle $	$\displaystyle $
$\displaystyle $	$\displaystyle \implies$	$\displaystyle $	$\displaystyle a b$	$\backslash$	$\displaystyle c$	$\displaystyle $	$\displaystyle $	$\displaystyle $

$\blacksquare$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle a\)	\(\perp\)	\(\displaystyle b\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle \exists m, n \in \Z: m a + n b\)	\(=\)	\(\displaystyle 1\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Integer Combination of Coprime Integers
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle c m a + c n b\)	\(=\)	\(\displaystyle c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle b s m a + a r n b\)	\(=\)	\(\displaystyle c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle a b \left({s m + r n}\right)\)	\(=\)	\(\displaystyle c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
\(\displaystyle \)	\(\displaystyle \implies\)	\(\displaystyle \)	\(\displaystyle a b\)	\(\backslash\)	\(\displaystyle c\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)