gcd 0 0 = 0

Marc van Dongen dongen@cs.ucc.ie
Sun, 16 Dec 2001 13:35:59 +0000

Marc van Dongen (dongen@cs.ucc.ie) wrote:

:   An integer $a$ divides integer $b$ if there exists an integer
:   $c$ such that $a c= b$.


:  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.


Marc van Dongen