GCD of Integer and its Negative

Theorem
Let $a \in \Z$ be an integer.

Then:
 * $\gcd \set {a, -a} = \size a$

where:
 * $\gcd$ denotes greatest common divisor
 * $\size a$ denotes the absolute value of $a$.

Proof
From Integer Divisor Results, the divisors of $a$ include $a$ itself.

From Integer Divides its Negative, $a \divides \paren {-a}$.

Thus we have:
 * $a \divides a$

and:
 * $a \divides -a$

and so:
 * $\gcd \set {a, -a} \ge \size a$

From Absolute Value of Integer is not less than Divisors, there is no divisor of $a$ which is greater than $a$.

That is:
 * $\gcd \set {a, -a} \le \size a$

Hence the result.