# Talk:Greatest common divisor

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 Definition:  The largest positive natural number which divides evenly all numbers given. [d] [e]

## Example is redundant

Oops, maybe I shouldn't have put in an example of Euclid's algorithm, since such an example is already given on the Euclid's algorithm page. --Catherine Woodgold 08:38, 13 May 2007 (CDT)

## Why so complicate?

$60 = 2^2 \times 3^1 \times 5^1$
$72 = 2^3 \times 3^2 \times 5^0$

So for the gcd you have take take the smallest exponents: :$\operatorname{gcd}(60,72) = 2^2 \times 3^1 \times 5^0 = 4 \times 3 = 12$

lcm is similar: You have to take the gratest exponents: :$\operatorname{lcm}(60,72) = 2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360$

--arbol01 19:01, 15 July 2007 (CDT)

That's what the article says. Are you suggesting that there's some difference between what you wrote above and what the article says? Michael Hardy 09:38, 16 July 2007 (CDT)

## highest common factor?

In number theory, I never read the term "highest common factor", but my Oxford dictionary and google seem to know it quite well. Is this perhaps a term used at school level? Peter Schmitt 23:41, 26 June 2009 (UTC)

It was the only term used when I was at school in the sixties. Ro Thorpe 18:20, 20 April 2010 (UTC)

## Subpage "Examples" or "Tutorial"?

The detailed examples should go on a subpage (Example, Tutorial?). Or is this what is meant by "Student level"? Then the name is a bad choice (at least for mathematics). Peter Schmitt 23:07, 27 June 2009 (UTC)

I just saw that there is a "Tutorials" subpage. That seems to fit in this case. In other cases "Example(s)" would be better. Peter Schmitt 23:53, 29 June 2009 (UTC)

Tutorial pages were added to Citizendium between the last revision before yours and your latest revisions. I'm not sure what their specific purpose is. The tutorial page doesn't give any description standardizing their purpose. Perhaps you are right that for a topic such is this, it should provide more extensive examples.
However, I don't really like the new version better than the old. Certainly some things are good additions -- mentioning relatively prime, for instance. But I don't think the existence of a tutorials page should preclude including an example or two on the main page. I like the old example better then the ones you added, both for formatting and content. The new examples are hard to read. They also involve applying a theorem, namely, that to compute a GCD, you can first factor (which requires knowing about the Unique Factorization theorem), then use the largest of the exponents common to each number to form the factorization of the GCD. I think the description of finding the GCD by enumeration as was originally done is conceptually simpler and doesn't require the uninformed reader to consult another page.
Even if there is consensus that the new type of example is preferable, the statement of the theorem that is used in the computation comes after the example itself. This is a particular instance of bad organization in the current revision. There is no introduction, and the topics seem haphazardly thrown together. Are there any thoughts from others on whether to modify the current version, combine it with the old, or to revert to the old completely with a few necessary additions (relatively prime, pairwise relatively prime, alternate definitions (or characterizations?) of GCD...)? Also, does anyone know of a post somewhere discussing the specific intention of a tutorials page?Barry R. Smith 04:06, 22 July 2009 (UTC)

## Clarifying wording

Suppose the two numbers being examined are 12 and 24. What's the greatest common divisor? 12. It's not less than 12. It's not between 1 and 12. It IS 12. The wording the greatest common divisor of some numbers is a number between 1 and the smallest of the numbers suggests that the number is between 1 and the smaller of the numbers, namely 12. The gcd IS 12. The revised wording is right.--Thomas Wright Sulcer 02:32, 20 April 2010 (UTC)

Yes, but 1 is also possible. Boris Tsirelson 07:23, 20 April 2010 (UTC)
I would say that between is ambiguous, but not incorrect. As a non-expert in this field, I would say that your current solution "the greatest common divisor of any numbers is a number greater than or equal to 1 and less than or equal to the smallest of the numbers" is one heck of a mouthful. I would suggest that a more readable say of putting this would be "the greatest common divisor of some numbers is a number between 1 and the smallest of the numbers inclusive", which clears up any ambiguity. However, I have no expertise in this field so I shall not make the edit. --Chris Key 10:44, 20 April 2010 (UTC)
Fair enough. I thought 1 had been excluded as a g.c.d. but clearly 1 goes into every whole number, and if the general consensus is to include 1, then I think the new wording is just fine. But I like the wording by Boris Tsirelson much better. I also like the Chris Key "inclusive" wording too. But don't we all agree that the new wordings (by BT and CK) are better? What I'm saying is that there's a benefit to this give-and-take, that sometimes conflict forces thinking, and that out of the conflict, the ideas that emerge are tougher, sharper, better understood, clearer, better explained. If I refuse to challenge a given wording because "I'm not an expert in the field" but I still think it's incorrect or ambiguous, and if I had remained mute about this, since I'm only a handyman (although I depend on accurate numbers to build walls straight etc), then the whole project loses.--Thomas Wright Sulcer 11:28, 20 April 2010 (UTC)
The question whether 1 is to include is not a matter of "consensus". If it were to be excluded the whole article would need to be changed.
As Chris says the explicit
"the greatest common divisor of any numbers is a number greater than or equal to 1 and less than or equal to the smallest of the numbers"
is rather long and not inviting for non-mathematicians. Thus I think that -- in particular in the introduction -- a simple language should be used if at all possible. Now, in my understanding, if I ask someone to choose a number between 1 and 100 the bound will be naturally included. Therefore, I still believe that it is not ambiguous (in this context), even more since the bounds have been explicitly mentioned as possible values in the same sentence. But if it is ambiguous for a non-mathematician then, for readability, one of the following versions should be used:
(1) "the greatest common divisor of some numbers is a number between 1 and the smallest of the numbers inclusive"
(2) "the greatest common divisor of some numbers is a number between 1 and the smallest of the numbers (bounds included)"
--Peter Schmitt 16:01, 20 April 2010 (UTC)
I like (1) better than (2). But I like Boris's wording too. My whole problem was the word between suggested 1 < x < smallest number. But what I think we want is 1 (< or =) x (< or =) smallest number, right? I don't think there's a way to say this cleanly, but it would be fun to fuss about this as if we were in a Monty Python skit, but then others might wonder whether some kind of bromantic relationship was forming.--Thomas Wright Sulcer 16:25, 20 April 2010 (UTC)
Nº 1 looks good to me. Ro Thorpe 18:27, 20 April 2010 (UTC)

(unindent) Thanks, Ro. I have changed it to (1). But now I wonder if this is still ambiguous because it is not clear if "inclusive both numbers"?
In (2) I should have used "endpoints" instead of bounds. This would certainly remove all ambiguity.
While I agree that amibiguities have to be avoided (even potential ones), I was curious end checked out "between": If you search for "number between" you will only find examples where the endpoints are assumed to be included. This satisfies me because it shows that my knowledge of colloquial/common English is not as bad as the comments seemed to indicate. --Peter Schmitt 23:17, 20 April 2010 (UTC)

Absolutely no problem with your English: 'inclusive' automatically includes both endpoints (yes, itself better than the rather unidiomatic 'bounds'). As for 'between', 'zwischen', 'entre', I think it's a concept that survives translation - though I'm happy to see you are satified that in a scientific context it too includes the endpoints. Ro Thorpe 00:02, 21 April 2010 (UTC)

## Grammatically speaking

The second sentence of the lead-in paragraph now reads: ".... the greatest common divisor of any numbers is a ....". Should it not be "any number" instead of "any numbers"? Or perhaps "any set of numbers" instead of "any numbers"? Milton Beychok 16:10, 20 April 2010 (UTC)

I agree Milton.--Thomas Wright Sulcer 16:28, 20 April 2010 (UTC)
"any number" would be grammatically correct, but mathematically wrong. It was "some numbers" two days ago. --Peter Schmitt 16:45, 20 April 2010 (UTC)
Fair enough, Peter ... "some numbers" would fix the grammar. As for the math, I leave that to you. Milton Beychok 17:06, 20 April 2010 (UTC)

## The "Examples" section lacks clarity for a non-mathematician

I must confess that the examples section was completely incomprehensible to me. Then I read the "Tutorials" sub-page and it was very clear and easily understandable. Why not include this just below the section header (indented as I have shown):

For more explanation, click the Tutorial tab at the page top.

Also, are those large dashes (—) needed in the next-to-last line of the "Examples"? They confused me no end. Milton Beychok 18:04, 20 April 2010 (UTC)

Thanks for your comments, Milton. Non-specialist views are always important. I'll take a look at them. I thought that they are simple enough: Since the numbers are very small it is easy to check the values.
What, in particular, was incomprehensible? What the examples are meant to show? The statements? The wording? The notation?
Shouldn't the links on the subpages template replace the necessity of links from the text? The link to the Tutorials subpage is clearly shown there.
--Peter Schmitt 23:26, 20 April 2010 (UTC)
Peter, the main article page has no obvious wiki links to the Tutorials subpage. The only pointer is the Tutorial tab at the top of the page and most new readers will not go to that tutorial before they read the article ... many may not go there even after reading the article. Therefore, I thought it would be useful to provide a pointer right at the beginning of the section just as we often do with the "Main" template.
As for what I found incomprehensible, looking at the second example:
"for 72 = 2 • 2 • 2 • 3 • 3, 108 = 2 • 2 • 3 • 3 • 3 there is (72,108) = 2 • 2 • 3 • 3 = 36"
I did not understand how the conclusion "there is (72,108) = 2 • 2 • 3 • 3 = 36" was reached. Obviously, I could see that it was correct, but how was it reached? I had the same problem with the rest of the examples as well. Also, I could not understand the use of the large dashes (—) ... are they meant to replace semi-colons (;) or what?
By contrast (at least for me), the tutorial was excellent and the explanations were very clear and very lucid. Without meaning to be egotistic, if I had trouble understanding the examples, you can be sure that most non-mathematicians will have the same trouble. Milton Beychok 00:11, 21 April 2010 (UTC)
It was probably a bad idea to put the examples before the methods, and to present the examples as briefly as possible. While I think that it is important to present the information -- at least on the top level -- in a concise way (in order to allow quick orientation, and to avoid "hiding" the important message behind too many words: long texts may not be read, or be only browsed) there is, of course, the danger of being too terse. Thus readability checks by others are needed. (I hope that the page is now readable.) Thanks, Milton.
If the tabs provided by the subpages template are not suggestive enough, then perhaps -- some time in the future -- we should try to revise it? --Peter Schmitt 10:18, 21 April 2010 (UTC)
The readability is now very much clearer, Peter. Thanks a lot. Milton Beychok 15:29, 21 April 2010 (UTC)