Divisor of Sum of Coprime Integers

Theorem
Let $a, b, c \in \Z_{>0}$ such that:
 * $a \perp b$ and $c \mathrel \backslash \left({a + b}\right)$.

where:
 * $a \perp b$ denotes $a$ and $b$ are coprime
 * $c \mathrel \backslash \left({a + b}\right)$ denotes that $c$ is a divisor of $a + b$.

Then $a \perp c$ and $b \perp c$.

That is, a divisor of the sum of two coprime integers is coprime to both.

Proof
Let $d \in \Z_{>0}: d \mathrel \backslash c \land d \mathrel \backslash a$.

Then:

A similar argument shows that if $d \mathrel \backslash c \land d \mathrel \backslash b$ then $d \mathrel \backslash a$.

It follows that:
 * $\gcd \left\{{a, c}\right\} = \gcd \left\{{b, c}\right\} = 1$

Hence the result.