Divisor of Sum of Coprime Integers

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

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

Then:

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

It follows that $\gcd \left\{{a, c}\right\} = \gcd \left\{{b, c}\right\} = 1$, hence the result.