GCD with Zero

Theorem

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

Then:

$\gcd \set {a, 0} = \size a$

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


Proof

Follows from:

Integer Divides Zero

and:

GCD for Negative Integers.

$\blacksquare$


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 $2.2$: $10$
  • 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $13$
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