# gcd 0 0 = 0

Marc van Dongen dongen@cs.ucc.ie
Tue, 18 Dec 2001 09:32:49 +0000

```Alan Bawden (Alan@LCS.MIT.EDU) wrote:

:    In case it isn't clear already, these definitions make a lattice on
:    the positive integers, with divides ~ leq, gcd ~ meet and lcm ~ join,
:    using the report's definitions of gcd and lcm.
:
: Indeed, that's a nice way of putting it.  How about if the report just
: says:
:
:    In order to make the non-negative integers into a lattice under `gcd'
:    and `lcm', we define `gcd 0 0 = 0'.

It would surely make things a lot less accessible to people (including
me) who do not have any (or limited) knowledge about lattices. Why not
make it more accessible and use the following rule (ore something similar)?
The greates common divison (gcd) of two integers a and b is the unique
non-negative integer g which has each of the following two properties:
1) g divides both a and b; and
2) if g' also divides both a and b then g' also divides g,
Here an integer a divides an integer b if there is an integer c
such that b = c*a.

Note that if you regard an integer a to be greater than another
integer b if b divides a then the gcd of two intgerers may also
be regarded as the greatest common divisor of a and b.

Regards,

Marc van Dongen

```