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Is the definition of divisor truly strictly as an integer in all math systems and practical applications of math?

I would have thought the term in more general use would be almost synonimous with denominator.

In other words it would not bother me unduly if an engineer referred to the denominator in the expressions 3/3.14 or 3/pi or (3x+1)/(x+1) as a divisor.

Would this be completely wrong? user:mirwin

I wrote this generalization:

"Let C, D and B be natural numbers, being B>D, Let C digits be written in a B base. For any D that complies with (B mod D = 1), if the D modulus of the sumatorium of C digits is 0, this implies that C mod D = 0."

...but I am not a professional mathematician. This theorem is likely to be hundreds of years old. ¿Can some expert tell me who discovered this, or if this is considered "common knowledge" or something. I would be most grateful.


Vicente Aceituno

I don't know who discovered it, but I'm sure it's as old as modular arithmetic. In your statement, you can safely drop the assumption B>D, and you can sharpen it a bit by stating: (sum of digits of C) mod D = C mod D. In particular, the sum of digits of C is divisible by D if and only if C is divisible by D. The proof goes like this: if B ≡ 1 (mod D), then Bj ≡ 1j = 1 (mod D), and so

AxelBoldt 18:58, 17 Sep 2004 (UTC)

Thanks a lot. I am not sure if the sharpening breaks something, as what I wanted to do was to explain why divisibility rules work. Particularly the division by 3 rule, that tantalized me since I was a kid. vaceituno 12:0, 19 Sep 2004 (UTC)

Dividing by 13

Article currently says, "A number is divisible by 13 iff the result of adding 4 times the last digit to the original number" ... which doesn't look finished. How should it end?

ZagrebMike 23:15, 22 Nov 2004 (UTC)

Yes, i agree. Who can explain?

...Nobody know

I know, I think. 4 is the multiplicative inverse of 10 mod 13. So (10a + b) | 13 iff (40a + 4b) | 13 iff (a + 4b) | 13. So for example you can reduce determining if 12056629 divides 13 to determining if 1205662 + 4*9 = 1205698 divides 13. — ciphergoth 17:40, July 24, 2005 (UTC)

However, -9 == 4 mod 13 so the existing rule also works, and multiplying by 9 is probably easier than multiplying by 4 - just calculate 1205662 - 90 + 9. — ciphergoth 17:47, July 24, 2005 (UTC)
A Vedic algorithm says that since a divisor of 13 times 3 gives 39. 39 is in the nines family so we add one and drop the units digit. Hence, the Ekādhika or multiplier is 4.

Example: Is 12,056,629 divisible by 13? Start on the right, move column-wise to the left. Multiply the first digit by 4, add the product to the next digit to the left, repeat the process on this result and add that result to the next digit to the left. If you end with either 13 or zero, then, yes, the number is divisible by 13, otherwise, no, not divisible. This method is called digit-wise osculation. It follows the vedic ideal of one-line notation. The lower line is the whole necessary notation. Multipling the units digit by 4 and adding the tens and the next digit is mental math.

1   2   0   5   6   6   2   9 
13  3   10  22  14  41  38   

Larry R. Holmgren 03:28, 1 March 2007 (UTC)


From article:

...and every integer is a divisor of 0.

Does 0 divide itself?

no it doesn't, because any division by 0 is undefined Xrchz

No, 0 does divide itself. a|b <==> exists k: ka = b. Obviously for any k, k0 = 0, so the condition is satisfied. — ciphergoth 08:11, August 2, 2005 (UTC)
Yeah, but the problem is the "any k" part. If 0|0, then what does 0 ÷ 0 come out to? You can't make a case for any particular answer, so it's considered "indeterminate." (Division by 0 is indeterminate for 0, and undefined for any other number -- either way, it's not allowed.) Perhaps the definition should properly read: "We say m|n for any integers m and n iff there exists a unique integer k such that n = km." Just to clear it up. --Jay (Histrion) 19:53, 4 August 2005 (UTC)
Sadly, I can't find my copy of "the theory of numbers" at the moment, but I'm pretty sure that my definition of divisibility is the one they give, not yours. Using yours would make proofs much longer, because you'd have to prove that k was unique all the time. And it's sad to have an exception to the rule "every integer divides zero". It's better to allow an anomaly in the relationship between a|b and b &divides; a. — ciphergoth 06:57, August 5, 2005 (UTC)
I see your point. --Jay (Histrion) 18:28, 6 August 2005 (UTC)
I made a survey of mathematics textbooks in my college library. Including Knuth's Concrete Mathematics and Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers. The only one I could find that permitted 0 divides 0 was Ross and Wright's Discrete Mathematics. However, they then added a note in every proof and theorem that m could not be zero. In short the convention is that 0 does not divide 0 and that is what the article should state.
This issue is similar to wheather 1 is prime. In many imprecise definitions of primality it is--and historically it was considered to be--but to define 1 to be prime is not useful and force you to add an exception for it to almost every theorem involving prime numbers.
If no one has further information on this topic I will edit the definition accordingly. --Elfan 02:26, 7 March 2006 (UTC)
I shall have to find my copy of Niven and Zuckerman's "The Theory of Numbers" to contradict you on this. Suffice to say at this stage that the article is now inconsistent with itself, since the definition has not been changed. You'll need to update the definition to reflect this convention... — ciphergoth 16:26, 24 March 2006 (UTC)
Fixed consistency. Do Niven and Zuckerman actually use 0|0 anywhere? --Elfan 18:19, 26 March 2006 (UTC)

It needs to be mentioned that although 0 is a factor of itsef, it is NOT a divisor of itself. —Preceding unsigned comment added by (talk) 10:34, 7 October 2009 (UTC)

It seems strange to me to agree that 0 is no divisor of 0 (as to achieve this 0 must be explicitly excluded in the definition), and in my study of mathematics in Belgium 0 was always a divisor of 0. But of course other agreements can be made, however there are some problems here. Namely the set of non-negative integers with the relation of divisibility, mentioned in this article, is now no partially ordered set (and no Lattice (order). Also as already mentioned in the paragraph 'Two definitions of gcd', the definition of greatest common divisor doesn't fit with this 'other' concept of gcd. In my country we also use 'a greatest common divisor' in a more general way where this is non-unique (so 'a' greatest common divisor in stead of 'the' greatest common divisor). For integers, -gcd(a,b) is then also 'a greatest common divisor' of a and b. When talking about 'the greatest common divisor' of course the unique positive greatest common divisor is meant then. Samaritaan (talk) 22:14, 21 October 2011 (UTC)


Being a Dane, I wonder if(f) this is a word missing in my vocabulary :-) [[1]]

Shall we say iff. It is math lingo. Oleg Alexandrov 15:09, 22 Apr 2005 (UTC)

Change of wording

In the rule to find n mod 1001 in order to test wheter n is a multiple of 7, or 13, I had written "divide the number in groups of three digits from right to left". It was changed to "left to right", which isn't what I meant. Dividing 1048576 in groups of three digits from left to right would yield 104'857'6, wouldn't it? Army1987 14:38, 2 May 2005 (UTC)

Divisibility rule

should I think be reinstated as a separate article - it's not central to the concept of "divisor" and there is enough material for a separate article. — ciphergoth 20:13, July 24, 2005 (UTC)

  • I'd certainly vote for that. --Jay (Histrion) 19:33, 4 August 2005 (UTC)
  • my vote goes for that too -- Yannick Gingras 17:36, August 6, 2005 (UTC)
  • Yes please. The way it stands now, this takes way too much space in the article, and now with the proofs thrown in. By the way, according to Wikipedia:WikiProject_Mathematics/Proofs proofs should not be that proeminent in articles, especially if they are long, as the recently added proof in the divisor article. Oleg Alexandrov 17:52, 6 August 2005 (UTC)
  • I agree - a new article "Divisibilty rules", and link to that. Bubba73 18:14, August 6, 2005 (UTC)
For some reason, though, I feel "rules" is the wrong term. Would "checks," "shortcuts," or "tricks" be better? Or maybe "criteria," which has a meaning closer to "rules" but seems a bit more appropriate? --Jay (Histrion) 20:08, 6 August 2005 (UTC)
  • OK, I moved the material to Divisibility rule, although I'd still like to talk about a name change for the "new" topic. --Jay (Histrion) 20:23, 6 August 2005 (UTC)

I just found this page. I read a book, Vedic Mathematics, 1965, 1978. In chapters 29 and 30, pages 273-295, the author gives an algorithm for divisibility called osculation. It gets complicated as there are several methods: digit-wise, whole number, positive and negative, complex, and multiplex osculation. It can be done mentally or with some notation of the results of an addition and multiplication. It does not require division! The positive osculator plus the negative osculator equals the divisor, P+Q=D. Example: is the number 5,293,240,096 divisible by 139? Round up to fourteen tens. The osculator (multiplier) is 14. Start on the right, moving column-wise to the left. 14x6=86. Then adding to the tens column gives 86+9=93. Osculate 93. 3x14+9+0=51. Next column, osculating 51 gives 1x14+5+0=19. Osculating 19 gives 14x9+1+4=131. Then osculate 131, 14x1+13+2=29. Next osculate 29, 14x9+2+3=131. Next osculate 131, 1x14+13+9=36. Osculate 36, 6x14+3+2=89. Osculate 89, 9x14+8+5=139. Yes, the number is divisible by 139. When we end on either zero or the divisor 139 (or a multiple of 139), then, yes, the number is divisible by 139.

5    2   9   3    2   4    0   0   9   6 
139  89  36  131  29  131  19  51  93

More later. Larry R. Holmgren 03:08, 1 March 2007 (UTC)


Looks like some technical problem in red text...I have no idea what to do about it. Gert2 03:35, 4 January 2007 (UTC)

Nevermind...I fixed it. Gert2 03:39, 4 January 2007 (UTC)

Possible error regarding abundant / deficient numbers

In the section "Further notions and facts" is this text:

"If a number equals the sum of its proper divisors, it is said to be a perfect number. Numbers less than the sum of their proper divisors are said to be abundant; while numbers greater than that sum are said to be deficient."

Is that right? I thought it was the other way around. Spadewarrior (talk) 00:39, 6 August 2008 (UTC)

I dont know the definition!!!!! sry!!!!! —Preceding unsigned comment added by (talk) 21:51, 21 January 2009 (UTC)


In the article, there are several times when the phrase "we say..." is used. I think it needs to be changed. Anyone disagree? Alan16 talk 00:25, 29 March 2009 (UTC)

Two questions re factors

Is a factor the same thing as a proper divisor?

Is n a factor of itself? The first part of the article seems to allow that possibility.

Thanks, CBHA (talk) 07:33, 30 May 2009 (UTC)

A software

I create a software for this. Please, add it in external links —Preceding unsigned comment added by Thekiing (talkcontribs) 16:25, 31 October 2009

Replied on user's talk page. JamesBWatson (talk) 16:54, 31 October 2009 (UTC)


I came here for a quick reference to see if it's the one on the top, or the one on the bottom, and it doesn't tell you straight out at the top of the article. I think maybe at the top of the page there could be an image showing "dividend/divisor". Anyway, so the divisor is the one on the bottom. That's all I was making sure of. --Neptunerover (talk) 08:21, 25 January 2010 (UTC)

Two definitions of gcd

There seem to be two different definitions of gcd. According to the definition at Greatest common divisor, namely "gcd(x,y) = greatest n such that m*n = x for some m and m*n = y for some m", gcd(0,0) is undefined: setting m = 0 the equations hold for all integers n, and since the ordering on the integers has no greatest element, there is no greatest such n.

However, according to the definition in this article (infimum of the lattice of integers ordered by divisibility), gcd(0,0) is defined to be 0, the top element.

So while it is true, as this article says, that the meet is gcd, it is a different gcd than the one defined at Greatest common divisor, namely one extended to be total. Hairy Dude (talk) 21:55, 30 October 2010 (UTC)

Wrong link to Dutch Wikipedia

This is the current link: This should be the link: I would correct it if I knew how to do that, since I don't I hope that someone else does. The first word "deelbaar" is the translation of divisible, the second word "deler" is the translation of divisor. The Dutch Wikipedia has both entries while the Englisch Wikipedia doesn't have an entry for divisible. (talk) 15:20, 24 April 2011 (UTC)

d(n) is a multiplicative function??

It's written that:
The total number of positive divisors of n is a multiplicative function d(n) (e.g. d(42) = 8 = 2 x 2 x 2 = d(2) x d(3) x d(7))

But I don't think this is always true.

Let's take 12 whose divisors are 1, 2, 3, 4, 6 and 12. So d(12) = d(2 x 6) = 6.
But d(2) = 2 and d(6) = 4, so d(2) x d(6) = 8 !! (talk) 11:24, 2 August 2011 (UTC)

You didn't read the link to find the actual definition of multiplicative function, did you? f(ab)=f(a)f(b) is only required to be true when a and b are coprime. In your example, 2 and 6 are not coprime. —David Eppstein (talk) 15:53, 2 August 2011 (UTC)
One correct way to express the number of divisors of a number n is to express n in terms of its integer factorization. For example:
. — Glenn L (talk) 16:48, 2 August 2011 (UTC)

Straight-line "divides" symbol & absolute value symbol, aliquot, plus more on "wow" or why does my head hurt...worse?

Granted, my pre-frontal processing is partially impaired because of my sleepiness, but the language (not the terminology, but the explanations thereof) used in Wikipedia math articles hurts my head, even in this article. Like the writer of the earlier "wow" section, I find it hard to find and process information in this math article and others. I don't expect to understand even half the math in Wikipedia, but even the first half of this article made my brain backtrack.

I know this is not the Simple English Wikipedia, but I assume we would like to create articles or sections that open with information aimed at the novice but that grows in complexity until an expert would be satisfied. I don't know if I can help with this due to chronic illness, but I wanted to point it out.

I came to this article from the Aliquot disambiguation page. Why is aliquot used here? Its etymology or definition might help. As the text is, my mind searches for meaning and only finds, "Alice in Wonderland's kumquat?"

Also, my mind hit speed-bumps at the straight-line division symbol. That space in my head was already taken up by the absolute value symbol. A brief comparison might help. Oh, and there's two absolute value articles that have to do with math. I don't know if they are both needed. But one of them used a smaller in-line math font that required less of a transition in my mind for single-line equations.

For a lot of us, unfortunately, math eats up a lot of energy to process. Smoother transitions between levels of complexity, conceptual connections, and even fonts can encourage us to process a little further. It's tough going when you're already at a deficit in energy when you haven't even hit the core of the topic.

--Geekdiva (talk) 12:07, 17 March 2012 (UTC)

Divisor defined as denominator

I have added a second definition to the lead, covering the sense of "divisor" to as at the start of the Terminology section. I don't actually agree with my own addition: an article should be about a concept, and should not cover two independent concepts that happen to share the same name. I think that this article should cover only the sense in abstract algebra (possibly restricted to integers), with a hatnote with a redirect to the other sense. To get rid of the alternative definition will involve removing a few lines only. Comment? — Quondum 22:53, 1 August 2013 (UTC)

Don't worry, I've removed my additional definition, and made some changes to clear up some of the confusion between the concepts in this article. — Quondum 00:04, 2 August 2013 (UTC)

0 again

The question of whether or not zero is a divisor of zero has reappeared at the Math Project Talk page. The question interested me enough to go scrambling through a number of texts (algebra, number theory, intro to proofs). I found both sides of the issue equally represented. About half define a | b only for nonzero a. About half of those that permit a to be zero make no statement about 0 | 0, while the rest are quite explicit about this being true. The authors ran the gamut and included some very respectable mathematicians. Some quick examples:

Herstein is of the nonzero a camp and states that if integer m ≠ 0 then m | 0.
Paley & Weichsel permit a = 0 and say "As a consequence of this definition, we see that any integer a divides 0. Indeed even 0 | 0."
Niven is of the nonzero a camp and says "It is understood that we never use 0 as the left member ... in a | b."
Dean of the nonzero camp says "However, by definition, zero divides no integer."
McCoy of the nonzero camp says (paraphrased) "We could just as well allow a to be zero in the above definition, but this case is unimportant and it is convenient to exclude it."

The one thing that I did not find was any statement about uniqueness in the definition of divisor. Personally I kind of like that wrinkle in the definition and it may appear in some education literature, but I don't have a citation for it.

The current definition on this page only reflects one point of view on the issue and probably should be expanded to give a broader perspective. Bill Cherowitzo (talk) 04:01, 2 August 2013 (UTC)

I'm a little confused by your statement [a]bout half of those that permit a to be zero make no statement about 0 | 0. Surely 0 doesn't divide any integer other than 0. Did you mean "permit b to be 0" rather than "permit a to be 0"? --Trovatore (talk) 06:37, 2 August 2013 (UTC)
Sorry, I should have been clearer. What I meant was that these authors did not make any explicit statement about 0 | 0, permitting it to be inferred from the definition without further comment. The others brought this point to the fore. Bill Cherowitzo (talk) 19:16, 2 August 2013 (UTC)
I see nothing strange about so making and others not making a statement about a being zero: that an author bothers to make a specific statement about this special case when it is clear from the definition is not significant. To distil Bill's finding: the field is split between allowing and disallowing zero as a divisor (which is different from what found at the Math Project page (though relative notability is always an issue).
How about this hypothesis: authors coming from the educational perspective and dealing only with integers might exclude zero as a divisor, because of potential confusion between the dual uses of the term "divisor" (one in abstract algebra, one in arithmetic), and authors in a more formal mathematical context, especially abstract algebraists, uniformly allow zero as a divisor.Only a hypothesis: not checked Does the exclusion simplify the statement of any theorems? What convenience is gained by excluding it? If my hypothesis is valid, there may be an argument to relegate mention of zero being excluded as a divisor to a discussion or footnote, but not to treat it as having weight. — Quondum 14:31, 2 August 2013 (UTC)
It didn't really split that way and my impression (non-scientific) was that the algebraists were more likely to not allow zero. The data is pretty fuzzy though and I wouldn't make any hard and fast calls from it. Bill Cherowitzo (talk) 19:16, 2 August 2013 (UTC)
I was wondering about McCoy's "convenient" as well. (McCoy who, BTW?) I don't really see anything convenient about excluding it, except possibly that it would let you identify "a | b" with "b/a is a well-defined integer", but really I don't see how that enables any further convenience beyond its own bare statement. --Trovatore (talk) 18:03, 2 August 2013 (UTC)
Neal H. McCoy, Fundamentals of Abstract Algebra (1972). One convenience is that if you exclude it from being a divisor, then you don't have to do it when you define zero divisors. I've seen a couple of other statements in these comments which support the convenience. The only advantage that I can see for having 0 be a divisor of 0 is that then you can say that every integer divides 0 instead of every nonzero integer divides 0. A rather small gain in my mind. Bill Cherowitzo (talk) 19:16, 2 August 2013 (UTC)
Zero divisor is not to be confused with a divisor of 0.
Definition of zero divisor in a ring: is a pair of zero divisors ( is a left-sided zero-divisor, is a right-sided zero-divisor) .
An integral domain is a commutative ring with no (pairs of) zero divisors. Divisibility is usually defined in an integral domain (sometimes with unity). In this case, it is nonsensical to define zero divisor in terms of divisibility. So McCoy must have defined divisibility differently. How?
Also, what are the "other statements in these comments which support the convenience"? (talk) 08:44, 3 August 2013 (UTC)
Well, no, there are various others. It makes the description of Euclid's algorithm cleaner, for one thing. It turns the natural numbers into a very nice lattice structure with 1 at the bottom and 0 at the top, and then gcd and lcm are just meet and join respectively, and you don't have to make special exceptions to explain that gcd(n,0)=n for every n including 0 (the correct choices for Euclid's algorithm). --Trovatore (talk) 19:26, 2 August 2013 (UTC)
Does the concept of "divides" not extend to more general rings (that include zero divisors) using the same definition? If one forms a ring as the direct product of several rings, then under homomorphisms, the exclusion must become pretty inconvenient. In particular, homomorphisms should preserve the "divides" property, whereas if we exclude zero, it wouldn't, because some homomorphisms will map some zero divisors onto zero. This reasoning is out of the air, so don't crucify me if I've got it wrong. But if I'm not off-beam, divisor does not behave nicely under generalization to more general rings unless we include zero. — Quondum 23:50, 2 August 2013 (UTC)
Gentlemen please. I am not arguing (pro or con) whether zero is a divisor of zero. In fact, I have not expressed my opinion on the matter. I made my initial comment because the discussion was getting to be lop-sided and I felt that some balance was needed. I shouldn't have to remind you that our job is not to improve, correct or beautify the mathematics we are discussing, but rather to report on what we find in the secondary sources without bias. The reality here is that the sources are split and whether zero is a divisor of zero seems to be author-dependent. This can not be ignored because one's favorite example has a pretty property only under one definition (and by the way, it is not universally held that 0 is a natural number, a key point in Trovatore's example), nor can it be dismissed as a fringe concept which messes up more advanced theory (see Divisor (ring theory)). We should strive to present (in this instance) both sides and not give undue importance to this rather trivial question. When a statement requires one or the other definition to be valid, it should be explicitly mentioned. This is not an onerous burden for an editor of a short encyclopedia article. Bill Cherowitzo (talk) 15:33, 3 August 2013 (UTC)
I generally agree that, given the sources mentioned, we probably do need to give a nod to both conventions. And this is beside the point, but no, the convention that 0 is a natural number is not actually essential to my example — you would just have to rephrase it as being about the "nonnegative integers". --Trovatore (talk) 21:00, 3 August 2013 (UTC)
I too agree with Bill that its seems clear that both conventions should be mentioned; the only potential discussion would now be around whether one was dominant. In the above side-track, it was clear to me that it was a side-track exploring the prettiness of the concepts, not the notability thereof (and I guess I was simply extending Trovatore's lattice concept in a way where one definition seems to be clearly preferable, but as noted, this is not relevant).

With such encouragement I have edited the page in line with my previous comments. As to my choice of order in the definition, this version required the fewest number of changes/comments in the rest of the article (3) and avoided a nasty one in the definition of trivial divisors. Bill Cherowitzo (talk) 20:01, 5 August 2013 (UTC)

A more subtle point: there is a difference between excluding 0 as a divisor and defining it as not dividing anything, and it would be helpful to distinguish these, not clear at present. It would need careful reading of the sources to see how each author regarded each case. The answer to the following question under the definition excluding zero should be clear: is 0∤0 true? (It is either true or undefined; I would hope that there will not be authors who consider it to be true.) — Quondum 10:19, 6 August 2013 (UTC)

A recent edit cleanly removed the exclusion of 0 as a divisor, but in so doing going against the consensus to represent both of the prevalent definitions. I do not wish to revert this, because I think it would make sense to reintroduce the exclusion as a modification of the less restrictive case (i.e make it the second definition). This will improve readability. It will also make dealing with my "subtle point" easier as a later caveat, rather than something extra to be "forgotten" when removing the zero exclusion. — Quondum 17:52, 27 October 2013 (UTC)