# 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:

 $\displaystyle d$ $\divides$ $\displaystyle \paren {a + b}$ as $c \divides \paren {a + b}$ $\displaystyle \leadsto \ \$ $\displaystyle d$ $\divides$ $\displaystyle \paren {a + b - a}$ $\displaystyle \leadsto \ \$ $\displaystyle d$ $\divides$ $\displaystyle b$ $\displaystyle \leadsto \ \$ $\displaystyle d$ $=$ $\displaystyle 1$ as $d \divides a$ and $d \divides b$ which are coprime

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.

$\blacksquare$