Product of Coprime Factors

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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:

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


By definition of divisor:

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


\(\ds a\) \(\perp\) \(\ds b\)
\(\ds \leadsto \ \ \) \(\ds \exists m, n \in \Z: \, \) \(\ds m a + n b\) \(=\) \(\ds 1\) Integer Combination of Coprime Integers
\(\ds \leadsto \ \ \) \(\ds c m a + c n b\) \(=\) \(\ds c\)
\(\ds \leadsto \ \ \) \(\ds b s m a + a r n b\) \(=\) \(\ds c\)
\(\ds \leadsto \ \ \) \(\ds a b \paren {s m + r n}\) \(=\) \(\ds c\)
\(\ds \leadsto \ \ \) \(\ds a b\) \(\divides\) \(\ds c\) Definition of Divisor of Integer