Divisor of Sum of Coprime Integers

Theorem
Let $a, b, c \in \Z_{>0}$ such that:
 * $a \perp b$ and $c \divides \paren {a + b}$.

where:
 * $a \perp b$ denotes $a$ and $b$ are coprime
 * $c \divides \paren {a + b}$ 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 $d \in \Z_{>0}: d \divides c \land d \divides a$.

Then:

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

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

Hence the result.