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• ## factor distinguished from divisor ; was [PrimeNumbers] Prime formula

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• ... Not for ordinary rational integers, where it is important and valuable to preserve the distinction. Factors always form a product. A mere divisor need
Oct 30, 2001 1 of 16
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Tom Hadley, thadley@..., writes:

> Factor and divisor mean the same thing.

Not for ordinary rational integers, where it is important and valuable
to preserve the distinction.

Factors always form a product. A mere divisor need not.

I prepared the appendix, infra, to try to deal with the many defective
definitions offered for perfect.

Cheers.

Walter Nissen

---

Mathematicians and science writers, please take note:

Don't be confused by ring theory. Over Z+, factors always form a
product. If the factors cited don't form a reasonable product, then
they are merely divisors, not factors. E.g., 1, 2, 4, 7, and 14 are
factors of 784, merely divisors of 28.

These terms are well-formed:
divisor, factor, divisors, factors, the greatest common divisor, the set
of divisors, the set of positive divisors, the set of proper divisors,
sum of the proper divisors, the set of prime factors, the unique set of
prime factors, a set of factors, the sum of all its proper divisors.

These terms are ill-formed:
the set of factors, sum of the factors.

These terms are incorrect:
the sum of all the divisors, the sum of all its factors.

Thus, a perfect is the sum of its proper divisors.

When the context is clear, mathematicians often use abbreviations which
would be objectionable outside the context. E.g., in extremis, "the
factors of" for "the unique set of prime factors of".

Related terms not addressed here: unit, irreducible, ideal.
• ... You re losing me. So what happens when you multiply divisors together - don t you get a product? Are you trying to say that when considering the composite
Oct 30, 2001 1 of 16
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On Tue, 30 October 2001, Walter Nissen wrote:
> Tom Hadley, thadley@..., writes:
>
> > Factor and divisor mean the same thing.
>
> Not for ordinary rational integers, where it is important and valuable
> to preserve the distinction.
>
> Factors always form a product. A mere divisor need not.

You're losing me. So what happens when you multiply divisors together - don't you get a product? Are you trying to say that when considering the composite c,
the /set of factors of c/ can be distinct from
a /set of divisors of c/ in that
the /set of factors of c/ when multiplied together give you c again, but a /set of divisors of c/ do not necessarily have this property.
This, however, is _not_ a property of /a factor/, or /a divisor/, but of /sets/ thereof.

However, in that context, the set of factors I view more as a /decomposition/ of c, which explicitly (to my ears) indicates the re-multiplication to reform the composite property.

Phil

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• ... To the best of my knowledge, which is admittedly incomplete and imperfect, Walter is the only person to draw this distinction. I know of no other accepted
Oct 31, 2001 1 of 16
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Walter "Humpty Dumpty" Nissen wrote:

> Not for ordinary rational integers, where it is important and valuable
> to preserve the distinction.

To the best of my knowledge, which is admittedly incomplete and
imperfect, Walter is the only person to draw this distinction. I know
of no other accepted body of opinion which agrees.

> Factors always form a product. A mere divisor need not.

I fail completely to understand these two sentences. The words
individually make sense but their juxtaposition confuses me. The
attempted clarification:

product. If the factors cited don't form a reasonable product, then
> they are merely divisors, not factors. E.g., 1, 2, 4, 7, and 14 are
> factors of 784, merely divisors of 28.

doesn't help. The word "reasonable" doesn't seem to make sense in this
context.

Let's try asking a few questions to see whether the answers are
illuminating.

Are 2 and 14 factors of 28?

Are 4 and 7 factors of 28?

Are 1, 4 and 7 factors of 28?

Are 14 and 28 divisors of 784, or are they factors of 784?

Paul
• ... AFAIK, a factor and a divisor mean the same thing. ... No, these numbers are also factors of 28: 1*28 = 28 so 1 and 28 are factors of 28 2*14 = 28 so 2 and
Oct 31, 2001 1 of 16
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At 07:18 AM 10/31/2001 -0800, Paul Leyland wrote:

>To the best of my knowledge, which is admittedly incomplete and
>imperfect, Walter is the only person to draw this distinction. I know
>of no other accepted body of opinion which agrees.

AFAIK, a factor and a divisor mean the same thing.

> product. If the factors cited don't form a reasonable product, then
> > they are merely divisors, not factors. E.g., 1, 2, 4, 7, and 14 are
> > factors of 784, merely divisors of 28.

No, these numbers are also factors of 28:
1*28 = 28 so 1 and 28 are factors of 28
2*14 = 28 so 2 and 14 are factors of 28
4*7 = 28 so 4 and 7 are factors of 28

+---------------------------------------------------------+
| Jud McCranie |
| |
| Programming Achieved with Structure, Clarity, And Logic |
+---------------------------------------------------------+
• Divisor: number by which dividend is to be divided; number that divides another without remainder. Factor: one of the numbers that make up a number or
Oct 31, 2001 1 of 16
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Divisor: number by which dividend is to be divided; number that divides
another without remainder.

Factor: one of the numbers that make up a number or expression by
multiplication.

[Concise Oxford English Dictionary]

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Paul Leyland [mailto:pleyland@...]
Sent: 31 October 2001 15:19
To: Walter Nissen; primenumbers@yahoogroups.com
Subject: RE: factor distinguished from divisor ; was [PrimeNumbers]
Prime formula

Walter "Humpty Dumpty" Nissen wrote:

> Not for ordinary rational integers, where it is important and valuable
> to preserve the distinction.

To the best of my knowledge, which is admittedly incomplete and
imperfect, Walter is the only person to draw this distinction. I know
of no other accepted body of opinion which agrees.

> Factors always form a product. A mere divisor need not.

I fail completely to understand these two sentences. The words
individually make sense but their juxtaposition confuses me. The
attempted clarification:

product. If the factors cited don't form a reasonable product, then
> they are merely divisors, not factors. E.g., 1, 2, 4, 7, and 14 are
> factors of 784, merely divisors of 28.

doesn't help. The word "reasonable" doesn't seem to make sense in this
context.

Let's try asking a few questions to see whether the answers are
illuminating.

Are 2 and 14 factors of 28?

Are 4 and 7 factors of 28?

Are 1, 4 and 7 factors of 28?

Are 14 and 28 divisors of 784, or are they factors of 784?

Paul

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• ... I think that definition of divisor is or divisor and dividend of doing a division. Concise Encyclopedia of Mathematics: divisor: a number d which divides N
Oct 31, 2001 1 of 16
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At 06:07 PM 10/31/2001 +0000, Jon Perry wrote:
>Divisor: number by which dividend is to be divided; number that divides
>another without remainder.
>
>Factor: one of the numbers that make up a number or expression by
>multiplication.
>
>[Concise Oxford English Dictionary]

I think that definition of divisor is or divisor and dividend of doing a
division.

Concise Encyclopedia of Mathematics:

divisor: a number d which divides N [evenly, w/o remainder], also called a
factor.

factor: can be integer, polynomial, etc. The book doesn't give a precise
definition, but it is clear that a factor is a divisor.

+---------------------------------------------------------+
| Jud McCranie |
| |
| Programming Achieved with Structure, Clarity, And Logic |
+---------------------------------------------------------+
• As they are vague, I think I ll stick with my factor is a prime, and divisor is the product of one or more primes. Jon Perry perry@globalnet.co.uk
Oct 31, 2001 1 of 16
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As they are vague, I think I'll stick with my factor is a prime, and divisor
is the product of one or more primes.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Jud McCranie [mailto:jud.mccranie@...]
Sent: 31 October 2001 19:10
To: Jon Perry
Cc: primenumbers@yahoogroups.com
Subject: RE: factor distinguished from divisor ; was [PrimeNumbers]
Prime formula

At 06:07 PM 10/31/2001 +0000, Jon Perry wrote:
>Divisor: number by which dividend is to be divided; number that divides
>another without remainder.
>
>Factor: one of the numbers that make up a number or expression by
>multiplication.
>
>[Concise Oxford English Dictionary]

I think that definition of divisor is or divisor and dividend of doing a
division.

Concise Encyclopedia of Mathematics:

divisor: a number d which divides N [evenly, w/o remainder], also called a
factor.

factor: can be integer, polynomial, etc. The book doesn't give a precise
definition, but it is clear that a factor is a divisor.

+---------------------------------------------------------+
| Jud McCranie |
| |
| Programming Achieved with Structure, Clarity, And Logic |
+---------------------------------------------------------+
• ... This website says they are the same. http://www.utm.edu/research/primes/glossary/Divisor.html In fact the entry for factor links to divisor .
Oct 31, 2001 1 of 16
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At 07:52 PM 10/31/2001 +0000, Jon Perry wrote:
>As they are vague, I think I'll stick with my factor is a prime, and divisor
>is the product of one or more primes.

This website says they are the same.

http://www.utm.edu/research/primes/glossary/Divisor.html

In fact the entry for "factor" links to "divisor".

+---------------------------------------------------------+
| Jud McCranie |
| |
| Programming Achieved with Structure, Clarity, And Logic |
+---------------------------------------------------------+
• The point is we have two words which mean exactly the same thing. I m not going to force anyway to accept my usage, but I can t see any harm in the splitting
Nov 1, 2001 1 of 16
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The point is we have two words which mean exactly the same thing. I'm not
going to force anyway to accept my usage, but I can't see any harm in the
splitting of the definitions. Helps me out anyway.

Jon Perry
perry@...
http://www.users.globalnet.co.uk/~perry
BrainBench MVP for HTML and JavaScript
http://www.brainbench.com

-----Original Message-----
From: Jud McCranie [mailto:jud.mccranie@...]
Sent: 31 October 2001 22:43
To: Jon Perry
Cc: primenumbers@yahoogroups.com
Subject: RE: factor distinguished from divisor ; was [PrimeNumbers]
Prime formula

At 07:52 PM 10/31/2001 +0000, Jon Perry wrote:
>As they are vague, I think I'll stick with my factor is a prime, and
divisor
>is the product of one or more primes.

This website says they are the same.

http://www.utm.edu/research/primes/glossary/Divisor.html

In fact the entry for "factor" links to "divisor".

+---------------------------------------------------------+
| Jud McCranie |
| |
| Programming Achieved with Structure, Clarity, And Logic |
+---------------------------------------------------------+

Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
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Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
• Thanks for your responses. I apologize for not quoting all of your relevant points below. ... Wow, and all along I thought the subject to be number theory,
Nov 3, 2001 1 of 16
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Thanks for your responses. I apologize for not quoting all of your
relevant points below.

> "Humpty Dumpty"

Wow, and all along I thought the subject to be number theory, not the
politics of personal destruction. 8->

Factorization is a process which produces factors. Any user would be
less than satisfied by a program which produces too many "factors" or
too few. A claim to have factors is a claim to have "soup for
Goldilocks", just the right number. Even so, unless the product is
important, such a claim may be somewhat superfluous. In which case,
"divisor" will do just fine, just as it does when you don't have the
"right" number.

The rest of this message is at serious risk of being "much ado about
nothing much more".

Whenever a factor or divisor appears, there is always a product about;
if not in the foreground, then lurking in the background. Any set of
naturals can form a product. Simply multiply them all together. Then
the elements of the set are the factors of the product. In determining
the perfectness of 28, that product is 784. But it is perhaps merely a
distraction. Also, every division implies a product. If d | D, then
that product is D = d * q .

> However, in that context, the set of factors I view more as a
> /decomposition/

Exactly. As you know, when applied to naturals, this process is usually
called factorization.

Every product is associated with at least one set of factors. This
begins with the closure axiom itself. Every element of such a set is a
factor.

> Are 2 and 14 factors of 28?
> Are 4 and 7 factors of 28?
> Are 1, 4 and 7 factors of 28?

Yes. Yes. Yes.

> Are 14 and 28 divisors of 784, or are they factors of 784?

Yes; no, not by themselves.

> a factor can be equal to zero

Again, as you know, not in Z+.

In Z+, you eliminate the associates problem and reduce the units problem
to the multiplicative identity. It's a pretty nice system. Especially
with the Fundamental Theorem, i.e., unique factorization.

> The word "reasonable" doesn't seem to make sense in this context.

I notice you don't suggest an improvement. The word "reasonable"
could easily be the weakest in my earlier message. By using
"reasonable", rather than, say, "specific" or "uniquely determined", I'm
leaning over backwards to avoid correcting an author who may be able to
cite some tenuous reference or connection to the product.

It's best not to be mesmerized by parallels like this:

If a * b = c , then a and b are factors of c.

If a * b = c , then a and b are divisors of c.

These theorems suggest the close relationship, but don't extinguish the
distinction.

If you think this subject hasn't been beaten to death already, and wish
to reply, I would ask that you please not bloat hundreds of mailboxes by
quoting this entire message. Anyone who wants to see it can readily
look in the message archive.

Cheers.

Walter Nissen
• Better get used to it in this group. Its worse than being a politician. ... From: Walter Nissen To: Cc: my
Nov 3, 2001 1 of 16
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Better get used to it in this group. Its
worse than being a politician.

----- Original Message -----
From: "Walter Nissen" <wnissen@...>
To: <primenumbers@yahoogroups.com>
Cc: "my e-mail address is" <wnissen@...>
Sent: Saturday, November 03, 2001 3:22 PM
Subject: [PrimeNumbers] Re: factor distinguished from divisor

> Thanks for your responses. I apologize for not quoting all of your
> relevant points below.
>
>
> > "Humpty Dumpty"
>
> Wow, and all along I thought the subject to be number theory, not the
> politics of personal destruction. 8->
>
>
> Factorization is a process which produces factors. Any user would be
> less than satisfied by a program which produces too many "factors" or
> too few. A claim to have factors is a claim to have "soup for
> Goldilocks", just the right number. Even so, unless the product is
> important, such a claim may be somewhat superfluous. In which case,
> "divisor" will do just fine, just as it does when you don't have the
> "right" number.
>
>
> The rest of this message is at serious risk of being "much ado about
> nothing much more".
>
>
> Whenever a factor or divisor appears, there is always a product about;
> if not in the foreground, then lurking in the background. Any set of
> naturals can form a product. Simply multiply them all together. Then
> the elements of the set are the factors of the product. In determining
> the perfectness of 28, that product is 784. But it is perhaps merely a
> distraction. Also, every division implies a product. If d | D, then
> that product is D = d * q .
>
>
> > However, in that context, the set of factors I view more as a
> > /decomposition/
>
> Exactly. As you know, when applied to naturals, this process is usually
> called factorization.
>
> Every product is associated with at least one set of factors. This
> begins with the closure axiom itself. Every element of such a set is a
> factor.
>
>
> > Are 2 and 14 factors of 28?
> > Are 4 and 7 factors of 28?
> > Are 1, 4 and 7 factors of 28?
>
> Yes. Yes. Yes.
>
> > Are 14 and 28 divisors of 784, or are they factors of 784?
>
> Yes; no, not by themselves.
>
>
> > a factor can be equal to zero
>
> Again, as you know, not in Z+.
>
> In Z+, you eliminate the associates problem and reduce the units problem
> to the multiplicative identity. It's a pretty nice system. Especially
> with the Fundamental Theorem, i.e., unique factorization.
>
>
> > The word "reasonable" doesn't seem to make sense in this context.
>
> I notice you don't suggest an improvement. The word "reasonable"
> could easily be the weakest in my earlier message. By using
> "reasonable", rather than, say, "specific" or "uniquely determined", I'm
> leaning over backwards to avoid correcting an author who may be able to
> cite some tenuous reference or connection to the product.
>
>
> It's best not to be mesmerized by parallels like this:
>
> If a * b = c , then a and b are factors of c.
>
> If a * b = c , then a and b are divisors of c.
>
> These theorems suggest the close relationship, but don't extinguish the
> distinction.
>
>
> If you think this subject hasn't been beaten to death already, and wish
> to reply, I would ask that you please not bloat hundreds of mailboxes by
> quoting this entire message. Anyone who wants to see it can readily
> look in the message archive.
>
>
> Cheers.
>
>
> Walter Nissen
>
>
> Unsubscribe by an email to: primenumbers-unsubscribe@egroups.com
> The Prime Pages : http://www.primepages.org
>
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
• ... Surely this is a stupid definition, you say 4 is not a factor of 36, but 4 and 9 are factors of 36. Surely you can see why that isn t reasonable? Michael.
Nov 3, 2001 1 of 16
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>
> > Are 2 and 14 factors of 28?
> > Are 4 and 7 factors of 28?
> > Are 1, 4 and 7 factors of 28?
>
> Yes. Yes. Yes.
>
> > Are 14 and 28 divisors of 784, or are they factors of 784?
>
> Yes; no, not by themselves.
>

Surely this is a stupid definition, you say 4 is not a factor of 36, but 4
and 9 are factors of 36. Surely you can see why that isn't reasonable?

Michael.
• ... By themselves ? Hmmm. Doesn t the fact that 28 is a divisor/factor of 784 imply that 784/28 is also a divisor/factor?
Nov 3, 2001 1 of 16
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At 06:22 PM 11/3/2001 -0500, Walter Nissen wrote:
> Are 14 and 28 divisors of 784, or are they factors of 784?

>Yes; no, not by themselves.

"By themselves"? Hmmm. Doesn't the fact that 28 is a divisor/factor of
784 imply that 784/28 is also a divisor/factor?

+---------------------------------------------------------+
| Jud McCranie |
| |
| Programming Achieved with Structure, Clarity, And Logic |
+---------------------------------------------------------+
• ... the ... Hmmm. Too subtle perhaps. I suggest that you consult the seminal works of that great 19th century mathematician Charles Dodgson for further
Nov 4, 2001 1 of 16
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> From: Milton Brown [mailto:miltbrown@...]
> Better get used to it in this group. Its
> worse than being a politician.
...
> From: "Walter Nissen" <wnissen@...>
...
> > Thanks for your responses. I apologize for not quoting all of your
> > relevant points below.
> >
> >
> > > "Humpty Dumpty"
> >
> > Wow, and all along I thought the subject to be number theory, not
the
> > politics of personal destruction. 8->

Hmmm. Too subtle perhaps. I suggest that you consult the seminal works
of that great 19th century mathematician Charles Dodgson for further
insights into what I meant by those words.

Paul

P.S. Don't take it too personally 8-)
• ... Here s the passage that Paul had in mind: There s glory for you! I don t know what you mean by glory , Alice said. I meant, there s a nice
Nov 4, 2001 1 of 16
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Paul Leyland wrote:

> I suggest that you consult the seminal works
> of that great 19th century mathematician Charles Dodgson
> for further insights into what I meant by those words.

Here's the passage that Paul had in mind:

"There's glory for you!"

"I don't know what you mean by 'glory'," Alice said.

"I meant, 'there's a nice knock-down argument for you!'"

"But 'glory' doesn't mean 'a nice knock-down argument',"
Alice objected.

"When I use a word,"
Humpty Dumpty said in a rather scornful tone,
"it means just what I choose it to mean -- neither more nor less."

Through the Looking-Glass (1872) ch. 6
• I m not sure why certain people have trouble with this concept. I really don t have an explanation for that. The only thing I can think of is ring theory,
Nov 11, 2001 1 of 16
View Source
I'm not sure why certain people have trouble with this concept. I
really don't have an explanation for that. The only thing I can think
of is ring theory, ideal theory.

Here is an analogy.

Take a typical flawed definition of perfect:
A perfect is the sum of all its factors.

Actually this isn't too typical.
Authors often account for one of these factors 8->
positive
proper

Let's account for both:
A perfect is the sum of its positive factors, except itself.

Take 28 as an example.
The "factors" are 1, 2, 4, 7 and 14.
Since factors imply a product, this product would be 784.
It's really hard to imagine how to obtain 28 as a product from that set
(though I confess 29 would be even more of a challenge 8-> ).
What would the rationale for selection be?

Now, by analogy, would you apply the same standard to divisor that you
apply to factor?

Suppose the set { 31 , 32 } is under discussion. Would you say the
elements of this set are integers? Would you say the elements of this
set are divisors? Of course, it _is_ a fact that they are divisors, of
say, 4063232. So what? How is that relevant to anything worth saying
here? Unless something more is happening, this is a set of integers.

And, I would suggest, in the context above, { 1 , 2 , 4 , 7 , 14 } is a
set of divisors.

Simple (albeit incomplete) operational process for authors:

If all numbers in a set are real, call them real.
If all are algebraic, call them algebraic.
If all rational, call them rational.
If all integer, call them integer.
If all divisors of some number under consideration, call them
divisors.
If all are factors of some number under consideration, call them
factors.
If all are prime factors of some number under consideration, call them
prime factors.

You can stop short in this process; you won't be incorrect. But you
also may not be making the best use of your communication. But going
too far in this process is wrong, misleading, confusing.

These distinctions are valuable and they should be preserved.

It's a simple matter of clarity.

> Doesn't the fact that 28 is a divisor/factor of
> 784 imply that 784/28 is also a divisor/factor?

Surely, yes, even 784\28. 8->

> > > a factor can be equal to zero

> > Again, as you know, not in Z+.

> Who did decide this restriction?

My remark, two messages ago, was so restricted. By me. I didn't want
to wander off into ring theory, and in a mailing list named
primenumbers, the naturals seemed like a natural 8-> set. Others are
welcome to take up Z, Q, etc.

> Of course, 'to be a factor' and 'to be a divisor' are not intrinsic
> properties of a number, they obviously depend on the context.
> Personally, I regard these words as the will of their author to give
> me more information (even fuzzy) about the origin, the destination,
> the possible use, whatever, concerning a number in a given context.

Yes, good.

> the meaning of a word is neither definitive nor the same for
> everybody.

Not all the money you receive is genuine. That doesn't mean the
counterfeit is less bad.

If we can agree on the meaning of terms such as "greater", "product",
"distinct", "odd", "set", "positive" (as distinguished from
"non-negative"), "integer", "monotonic", "prime", "factor", "twin
primes", "circle", "phi", then we can communicate more clearly and more
easily.

> Trying to legislate on that is not far from a pure waste of time.

Short of legislation is a process called standardization. It can be
formal or de facto. It's more of a good thing than a bad thing.

Cheers.

Walter Nissen
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