GCD of Sum and Difference of Coprime Integers

Theorem
Let $a, b \in \Z$ be coprime integers.

Then:
 * $\gcd \set {a + b, a - b} = 1 \text { or } 2$

where:
 * $\gcd$ denotes greatest common divisor.

Proof
Let:
 * $d = \gcd \set {a + b, a - b}$

We have:

Hence:
 * $1 \le d \le 2$

and the result follows.