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.

$\blacksquare$


Sources

  • 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 $2.2$: $10$
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