# Talk:Least common multiple

## Redirect from 'lowest common multiple'

I've heard this term describes as 'lowest common multiple'. maybe you could put in a redirect from that site to here? well i'm toddling off to bed now, goodnight. - Mark Ryan

There already was a redirect from Lowest_common_multiple. --Zundark, 2002 Jan 8

## Confusing Algorithm Section

The section explaining the algorithm doesn't make a lot of sense to me. Especially this part, "In other words, the least element is increased by the corresponding x whereas the rest of the elements pass from X(m) to X(m+1) unchanged." Could somebody add a specific example of how this algorithm is supposed to work? —Preceding unsigned comment added by 134.173.194.123 (talk) 17:50, 16 October 2009 (UTC)

## Terrible example

If a=0 and b=0 the lcm(a,b)=0?? That's a horribly stupid example.. Use a little creativity.. —Preceding unsigned comment added by 70.73.59.208 (talkcontribs)

I agree that it's incredibly stupid as an example, but I think it was intended as something other than an example, and the sentence was badly written. I've rephrased that sentence. Michael Hardy (talk) 23:39, 6 November 2008 (UTC)
I just inserted some citations and text to clarify that lcm(a,0) is nonstandard, since my textbooks indicate that it is meaningless. — Anita5192 (talk)
That's wrong; it's not nonstandard. Some authors define it and some don't. See the related discussion at talk:divisor#0 again. It's actually the most natural and useful definition (the more useful definition of a|b is that there is some natural number k (not necessarily unique) such that b=ka, and "greatest" and "least" should be interpreted in the partial order of divisibility, not the usual order on the naturals). --Trovatore (talk) 14:29, 15 October 2013 (UTC)

## Proposed text on calculation method

does anyone think that this is a better way of calculating the LCM of 2 #s than what's posted?

The formula that works best: We will use 12 and 5 for out examples. To find the LCM of two numbers, put the larger of the two over the smaller in a fration (larger=numerator, smaller=denominator (12/5). Then you simplify the numbers (if they can't be simplified, as in this case, put the same improper fraction next to the original), and put the simplified version next to the unsilmplified version. 12/5 12/5 Then you cross multiply. 12 x 5, 12x 5. (numerator of original times denominator of simplified version) In this case, the answer is 60. Developed by Ben Cook (Aged 12 at the time) with help from Rob Cook (Also see Renderman computer graphics program, developed in part by Rob Cook). The preceding unsigned comment was added by Beoknoc (talk • contribs) 20:53, 29 January 2006 .

That's equivalent to saying ${\displaystyle \operatorname {lcm} (a,b)={\frac {a\cdot b}{\operatorname {gcd} (a,b)}}}$, since simplifying fractions is equivalent to dividing both numerator and denominator by the gcd. --71.106.183.17 (talk) 04:39, 25 October 2008 (UTC)

ha ha this is a cool trick!

## section cleanup

I've put a "cleanup" tag on the section with the Venn diagram. The union of the two sets is naturally divided into three disjoint components: the intersection, and the two components of the symmetric difference. Only one of the latter two is non-empty. I'd like to see an example in which all three are non-empty. It's easy to come up with examples, but the graphics will require software that is not on the machine I'm at. Maybe I'll do something within the next couple of days. Michael Hardy (talk) 18:05, 18 February 2008 (UTC)

...and now I've replaced the image, rewritten the section, and deleted the "cleanup" tags. Michael Hardy (talk) 01:36, 19 February 2008 (UTC)

## Huh?

• The previous formula utilizes the greatest common divisor, which itself utilizes the least common multiple in its formula. To break out of this circle ...

Does anyone know what he's talking about?

Virginia-American (talk) 21:38, 6 August 2008 (UTC)

It's nonsense. It stated that prime factorization is needed. That's not true since Euclid's algorithm can be used. (There are also very inefficient methods that don't use prime factorization; Euclid's algorithm is efficient.) I think the formula may have been correct once one got past the fact that it was not clearly explained, but I doubt it has much value for computational purposes. Michael Hardy (talk) 02:18, 7 August 2008 (UTC)
I was stating that some sort of algorithmic or computational method (such as Euclid's algorithm or prime factorization) is needed, and should have stated that more clearly. In terms of the formulas - yes the formulas indeed work, and the way in which they were derived simply by representing numbers as angular velocities is quite interesting. But, I do realize that you were right in deleting it for the reasons you pointed out - reducing a fraction in lowest terms does wind up requiring the GCD, and I posted it before thoroughly analyzing it. Thank you for noting this and acting accordingly. There are a number of results related to the stated formula that point to reducing the fraction by use of other means, but I am not assured that these results are stated clearly or are even possible as of yet. If I find that the author does include some intelligible way of reducing to lowest terms without the already well known methods of doing so, then I will certainly post to the discussion page, but I am doubtful that they exist. Telltree (talk) 20:18, 23 September 2008 (UTC)

## Comment moved from article

'READ PLEASE PARENTS AND KIDS'ONe thing this doesnt make any sense so if tweens read this..... they probably wont even know what this means we need someone to write something that will make sense to kids. And most kids aren't geniuses(if you are a kid genius your cool) but as to everyother kid in the world THEY AREN'T GENIUSES.....So put some stuff in here that makes sense. —Preceding unsigned comment added by 148.177.1.210 (talkcontribs)

## Positive integers?

I think that the formula

${\displaystyle \operatorname {lcm} (a,b)={\frac {a\cdot b}{\operatorname {gcd} (a,b)}}}$

mentioned in the article is valid for positive integers only, isn't it? If I haven't overseen something, the article does not state this fact. --Kompik (talk) 10:51, 23 September 2008 (UTC)

I think it might be best to restrict the whole discussion to positive integers at the beginning and then add others as a sort of afterthought, since all the ideas are there when one deals only with positive integers. Michael Hardy (talk) 12:09, 23 September 2008 (UTC)

Is it ever extended to non-integers? The LCM(pi,e) would be pi*e, for instance, LCM(4*i,6*i) would be 12*i, LCM(2/3,7/6) would be 14/3. —Preceding unsigned comment added by 71.167.66.173 (talk) 00:36, 31 May 2009 (UTC)

If the LCM of π and e were just their product, then why introduce such a concept? We already have a concept of multiplication. If you mean the smallest integer multiple that those two numbers share in common, I suspect there isn't any. Such a thing would exist only if their quotient is rational, which seems improbable. Michael Hardy (talk) 03:01, 14 September 2009 (UTC)

## OED reference

There seems to be some contention over whether the OED is suitable as a reference for the lead of the article. Firstly, the citation to the OED does not support the lead as it is currently written, which is about rational numbers. Secondly, the OED is not an especially good source for mathematics (it is, after all, just a dictionary). A good source would be, for instance, the considerably authoritative "Introduction to the theory of numbers" by G. H. Hardy and E. M. Wright (which focuses on the case of integers exclusively). The tertiary source Encyclopedia of Mathematics also confines attention (at first) to integers: {{#invoke:citation/CS1|citation |CitationClass=citation }}. What mathematical sources follow the seemingly novel approach of the article? If such sources are not forthcoming, then I suggest that the lead should emphasize the integer case again, in the spirit of keeping a neutral point of view. Sławomir Biały (talk) 11:12, 14 September 2010 (UTC)

Also, here is the OED definition, for reference: "the smallest quantity that has two or more given quantities (and no others) as its factors." (One can, in any event, take issue with the accuracy of this statement, particularly if "quantity" is interpreted to mean something other than "integer".) Sławomir Biały (talk) 11:34, 14 September 2010 (UTC)

I have removed the references to the rational case. It has been more than a year since I asked for sources, and none were given. Sławomir Biały (talk) 11:52, 17 October 2011 (UTC)

## formulas

I added a section of formulas. Because of the duality, they belong as much in the article on gcd. Is there a way to share pages between articles? Virginia-American (talk) 21:38, 15 September 2010 (UTC)

I would als like an article like "Identities on GCD and LCM" analogously to, say List of trigonometric identities. I am still missing identity laws like "gcd(0,a) = a, lcm(1,a) = a". HenningThielemann (talk) 09:21, 18 February 2011 (UTC)

## Error

The "3" line in the image at the top of the page is missing 1 data point and the rest is shifted to the left.Psychic94 (talk) 03:13, 6 April 2013 (UTC)

- AGREE! I'll try to come up with a new diagram. 84.114.26.36 (talk) 01:11, 23 November 2013 (UTC)

## Algorithm section needs rewriting by English speaker

The explanation of the algorithm for computing the LCM doesn't define the notation it uses - the superscripts are normally exponentiation, but that makes no sense in this instance. If I didn't already understand what it was trying to say, it would make absolutely no sense to me. Also, the English is bizarre: "The purpose of the loop is to"? If the loop does X, just say that it does X. Saying instead that "the purpose is" implies that it was intended to do X, but fails to do so because it is buggy. Some of the cryptic notation and English is because this text was copied from the explanation by a non-native english speaker in one of the external links. This page needs to be re-written by a native (or competent non-native) English speaker. — Preceding unsigned comment added by 14.202.196.207 (talk) 15:51, 8 June 2013 (UTC)

The notation used in A simple algorithm is standard for algorithms. Nonetheless I have inserted a parenthetical remark to clarify the notation to those not already familiar with algorithms. Does this make it more clear to you? — Anita5192 (talk) 23:31, 8 June 2013 (UTC)

## Landau reference

In my copy of Landau (Chelsea NY) the three identities listed are on page 234 not 254. It could also be useful to list the part of the book, i.e. part 1, chapter III, exercise 3. 2A02:810B:80C0:294:3844:D9BF:C395:C2BE (talk) 20:09, 10 February 2014 (UTC)