GCD with Self

Theorem
Let $a \in \Z$ be an integer such that $a \ne 0$.

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

where $\gcd$ denotes greatest common divisor (GCD).

Proof
From Integer Divides its Absolute Value:
 * $\size a \divides a$

Then from Absolute Value of Integer is not less than Divisors:


 * $\forall x \in \Z: x \divides a \implies x \le \size a$

The result follows by definition of GCD.