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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.

Date: 07/03/2015 at 01:33:39
From: Danny
Subject: Mistake in least common multiple with zero.

Hi,

Twelve years on, I found this conversation. In it, Doctor Rick states 
that if we allow 0 to be the LCM of any non-zero number, then 0 will be 
the LCM of all numbers.

I am an undergraduate math major in an Australian university, so correct 
me if I am wrong, but I believe that statement isn't true.

Where a|c reads "a divides c," the definition of c = LCM{a,b} is:

    i) a|c and b|c
   ii) if a|d and b|d, then c|d

Now suppose that c = LCM{3,2} = 0. Then let d = 6. We have 3|d and 2|d. 
But 0 = c does not divide d, since no number m exists such that 
m|0 = 6 = d. So in fact, letting 0 be the LCM of a pair of numbers will 
lead to a contradiction in the usual setting. 

In the set N of natural numbers with zero, it can be proved from the 
definition of LCM that 0 is not the least common multiple of any finite 
subset of N, though zero is the least common multiple of N itself, since 
(i) holds for all elements of N and (ii) holds for d = 0 only.

I believe the author mixed up the usual order relation in the definition 
of LCM, thinking that a least common multiple must be less (in numerical 
value) than other common multiples. 


Date: 07/03/2015 at 09:54:02
From: Doctor Rick
Subject: Re: Mistake in least common multiple with zero.

Hi, Danny.

Did you observe that I gave a reference for the definition of LCM that I 
used in my argument? MathWorld's definition, as I quoted, is: 

   The least common multiple of two numbers a and b, variously 
   denoted LCM(a,b) ... is the smallest positive number m for 
   which there exist positive integers z_a and n_b such that 
   n_a*a = n_b*b = m.

You cite a different definition. So much depends on definitions! What is 
the source for yours? Is it truly a definition, or is it a theorem about 
the LCM? (What I talked about was counterfactual -- what would happen if 
we changed the definition of the LCM? -- and if the definition is 
changed, theorems based on the definition might no longer be valid.)

In the 2003 conversation in question, I talked about what happens if we 
count zero as a common multiple. You aren't really dealing with that 
argument at all; you're saying nothing about "common multiples" broadly, 
only about the LEAST common multiple, and going straight to your 
definition of LCM. I could, and did, do the same thing with my definition 
(and MathWorld's): since it says the LCM must be positive, obviously it 
can't be the case that the LCM of any two numbers is zero. Case closed.

What I tried to do in the first part of my article was to approach the 
question of why the LCM is defined as it is. So I started from the 
common-sense meaning of the phrase "least common multiple" as the smallest 
number that is a multiple of both a and b. You tell me that this is not 
true, which leaves me somewhat confused. Can you show me an example of two 
numbers whose LCM is not "less (in numerical value) than other common 
multiples"?

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


Date: 07/03/2015 at 20:25:24
From: Danny
Subject: Mistake in least common multiple with zero.

Hi,

Thank you very much for your reply.

I apologise, I had not realised that there was a different definition 
cited in your earlier conversation.

I believe that this definition of LCM is useful when dealing with group 
theory, ordered sets in general, and lattices. In particular, it remains 
defined in dealing with the case of infinite sets, and sets where the 
natural order has been changed or altered. 

Also the definition works in the case where LCM is required for two 
numbers, one of which is zero. Zero will be a common multiple of all 
pairs of numbers, so using divisibility as an order relation 0 becomes 
the top element of an infinite ordered set like the naturals with zero. 
This set then forms a complete lattice with bottom element 1 and top 
element 0. It is a lattice because every pair of elements has a least 
upper bound, and greatest lower bound; and it is complete because the 
entire set has a least upper bound, 0, and a greatest lower bound, 1. 

The LCM of, for example, 0 and 3, will be 0, while the LCM of 2 and 3 
will be 6. 

I suspect that the definition you give is a useful version that works 
well in most cases; however, perhaps the definition I gave is a way that 
the LCM can be more general. 

Could you give me an example of why your definition might be more useful?

Also the proofs that I have seen for existence of the LCM use the 
definition that I gave, and I'm yet to find one that uses the least 
positive number definition, though I wouldn't be surprised if one exists. 

Part (ii) of the definition is particularly useful in proving existence. 
Could you show me to an existence proof which uses your definition?

Thanks for responding. This is definitely helping me think about my 
course and consider the reasons for these definitions more carefully!


Date: 07/03/2015 at 20:52:14
From: Doctor Rick
Subject: Re: Mistake in least common multiple with zero.

Hi, Danny.

I cannot help you at the level you are discussing, that of generalizing 
the LCM from its original, more intuitive application. I will invite 
other math doctors to discuss this matter with you.

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


Date: 07/04/2015 at 03:06:28
From: Doctor Jacques
Subject: Re: Mistake in least common multiple with zero.

Hi Danny,

As requested by Doctor Rick, I will add my comments.

Your definition is absolutely correct; and, unlike the "simpler" 
definition, it works in any commutative ring (although, in general, LCMs 
need not exist or be unique). The definition is the dual of the general 
definition for GCD.

Note that the original article was about LCM(a,b) where either a or b is 
0; this is not the case in your example with LCM(3,2).

In fact, I would say that there is no problem in considering that 0 is a 
common multiple of a pair of integers: after all, it is a multiple of 
each of them.... The point is that it is not the least such multiple, 
where "least" must be understood with respect to the partial ordering 
induced by divisibility, since this is the meaning used implicitly in 
"least common multiple." The problem is maybe in the "implicit" aspect. 
In any case, I think you know all this.

The last question is why we have the "simpler" definition. It is already 
taught in elementary school, at a time where the general definition would 
be out of reach. Even at that stage, the definition can be very useful in 
practical applications (like adding fractions); the same is true for 
greatest common divisor (GCD). For many people, that is about all the 
mathematics they will need.

I think that, if you want to learn math seriously, you have first to 
unlearn many of the (false or incomplete) things you were taught in 
school, because it was not possible at that time to give you strictly 
correct and complete definitions.

Please feel free to write back if you require further assistance.

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


Date: 07/04/2015 at 10:56:54
From: Doctor Peterson
Subject: Re: Mistake in least common multiple with zero.

Hi, Danny.

I'd like to tie things up with a comment on the big picture.

It is quite common for a concept to start with a simple idea and a 
"naive" definition, and later be generalized. In this particular case, as 
has been mentioned, both definitions MUST continue in use in different 
contexts, because only the naive initial definition is understandable by 
most people who need the concept, while only the sophisticated general 
definition applies to cases beyond natural numbers.

Most online sources, including Wikipedia and MathWorld, give the 
definition applicable to natural numbers. This is the appropriate 
definition for use as the Least Common Denominator of fractions, since 
denominators can't be zero. It also fits the name: it is exactly what it 
says, the LEAST (positive integer) multiple of the given numbers. This 
definition is undoubtedly the source of the entire concept.

Some sources give that same definition, but then add that if one of the 
numbers is zero, the LCM is 0 (with or without explaining why this 
extension makes sense). This is probably the answer that should have been 
given to the original question (from a computer science context, just 
looking for a reasonable value to give in this case).

This elementary definition had to be extended in order to cover other 
contexts, as you noted. As Doctor Rick pointed out, simply applying the 
basic definition to such cases would not work. The definition you are 
using is the result of a search for an appropriate extension, and is 
based on a theorem that is true in the natural number context, and turns 
out to be usable as a definition in the general case. It certainly would 
not be appropriate to start with this as a definition in elementary 
grades, but it would be possible to make the transition before getting to 
abstract algebra -- though it would never be particularly helpful in 
understanding the concept in its everyday applications.

I had trouble searching for a source for your definition, since the 
elementary definition is overwhelmingly common. As I mentioned, Wikipedia 
gives the elementary definition, but adds

   https://en.wikipedia.org/wiki/Least_common_multiple

   Since division of integers by zero is undefined, this definition
   has meaning only if a and b are both different from zero. However,
   some authors define LCM(a,0) as 0 for all a, which is the result
   of taking the LCM to be the least upper bound in the lattice of
   divisibility.

This last thought leads to your definition, which is given at the bottom 
of the page in defining the lattice of divisibility:

   The least common multiple can be defined generally over 
   commutative rings as follows: 

   Let a and b be elements of a commutative ring R. A common 
   multiple of a and b is an element m of R such that both 
   a and b divide m (i.e., there exist elements x and y of R 
   such that ax = m and by = m). A least common multiple of 
   a and b is a common multiple that is minimal in the sense 
   that for any other common multiple n of a and b, 
   m divides n.

   In general, two elements in a commutative ring can have no 
   least common multiple or more than one. However, any two 
   least common multiples of the same pair of elements are 
   associates. In a unique factorization domain, any two 
   elements have a least common multiple. In a principal ideal 
   domain, the least common multiple of a and b can be 
   characterised as a generator of the intersection of the 
   ideals generated by a and b (the intersection of a 
   collection of ideals is always an ideal).

So, summing things up, the definition Doctor Rick gave is very useful in 
the everyday world; yours is useful in higher-level mathematics -- and 
both can peacefully coexist because they give the same result where both 
apply. Both are "the correct definition" within their own context.

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


Date: 07/04/2015 at 21:33:32
From: Danny
Subject: Thank you (Mistake in least common multiple with zero.)

Thanks, guys. I appreciate the help :)

There's a lot to think about ...
Associated Topics:
Elementary Definitions
Elementary Multiplication
Middle School Definitions

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