# gcd 0 0 = 0

**Marc van Dongen
**
[email protected]

*Sun, 16 Dec 2001 13:35:59 +0000*

Marc van Dongen ([email protected]) wrote:
:* An integer $a$ divides integer $b$ if there exists an integer
*:* $c$ such that $a c= b$.
*
[snip]
:* gcd 0 0 = 0; and
*:* gcd 0 0 /= error "Blah"
*
To make clear why $0$ (and not any other non-zero integer) is the
gcd of $0$ and $0$ I should have added that for the integer case
$g$ is called a greatest common divisor (gcd) of $a$ and $b$ if it
satifies each of the following two properties:
1) $g$ divides both $a$ and $b$;
2) if $g'$ is a common divisor of $a$ and $b$ then $g'$ divides $g$.
First notice that $0$ is a gcd of $0$ and $0$ because of the following:
*) $0$ divides $0$ (and divides $0$);
*) whenever $g'$ is an integer that divides $0$ and divides $0$
then $g'$ divides $0$.
Next notice that if $g$ is any non-zero integer then $g$ cannot be
a gcd of $0$ and $0$ because $0$ (a common divisor of $0$ and $0$)
does not divide $g$.
Finally, observe that this makes $0$ the unique gcd of $0$ and $0$.
:* The gcd of two integers is usually defined as a non-negative
*:* number to make it unique.
*
Regards,
Marc van Dongen