Least Common Multiple with Zero

Date: 03/26/2003 at 12:03:20
From: Pops
Subject: Least Common Multiple with zero

I'm trying to find any reference about the least common multiple of 
two numbers when one (or both) is zero.

Could you help me, please?

Date: 03/26/2003 at 15:46:31
From: Doctor Rick
Subject: Re: Least Common Multiple with zero

Hi, Pops.

No number except zero is a multiple of zero, because zero times 
anything is zero. The only multiple that, say, 0 and 5 have in common 
is 0. Thus, if the LCM of 0 and 5 exists at all, it must be 0.

We do not count zero as a common multiple. If we did, then zero would 
be the least common multiple of any two numbers (unless we also 
counted negative multiples, in which case there would be no least 
common multiple of any two numbers).

Either we make an exception in this case, so that the LCM of zero and 
any number is zero, or we make no exception, in which case the LCM of 
zero and any number does not exist. To me it makes more sense to say 
that the LCM is defined only for positive numbers.

See the definition of LCM here:

   Least Common Multiple - Eric Weisstein, World of Mathematics
   http://mathworld.wolfram.com/LeastCommonMultiple.html 

It says, "The least common multiple of two numbers a and b is the 
smallest number m for which there exist POSITIVE integers n_a and n_b 
such that n_a*a = n_b*b = m." [Emphasis is mine.] If the LCM of 0 and 
5 were 0, we'd have a = 0, b = 5, n_a = any number, and n_b = 0 - 
which is not a positive integer, so it fails this definition. Thus, 
while the definition does not explicitly say that the two numbers 
must be positive, this is implied by the definition.

I have to ask: Why do you care? Is there a context in which you need 
the LCM of zero and another number?

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 03/28/2003 at 03:34:33
From: Pops
Subject: Least Common Multiple with zero

I'm a computer science professor and I'm proposing to the students a 
program to obtain the LCM of two numbers. My aim is to explain the 
correct answers in all possible "legal" situations.

Thank you very much for your response.

Pops.