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.

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

Then:

$\ds d$

$\divides$

$\ds \paren {a + b}$

as $c \divides \paren {a + b}$

$\ds \leadsto \ \ $

$\ds d$

$\divides$

$\ds \paren {a + b - a}$

$\ds \leadsto \ \ $

$\ds d$

$\divides$

$\ds b$

$\ds \leadsto \ \ $

$\ds d$

$=$

$\ds 1$

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.

$\blacksquare$

1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $2$: Some Properties of $\Z$: Exercise $2.2$
1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $2$: Divisibility Theory in the Integers: $2.2$ The Greatest Common Divisor: Problems $2.2$: $16 \ \text {(d)}$