Divisor of Sum of Coprime Integers

Theorem
Let $$a, b, c \in \Z^*_+$$ 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 $$d \in \Z^*_+: d \backslash c \land d \backslash a$$.

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.