RE: Factors and Terms and proper divisors

[Joel Fritz <JFritz@xxxxxxxxxxxxxxxx>]

I've never paid much attention to the distinction between factors and
divisors. Sorry for the lack of precision.  

> -----Original Message-----
> From: Leif Svalgaard [mailto:leif@leif.org]
> Sent: Saturday, October 21, 2000 12:02 AM
> To: MIDRANGE-L@midrange.com
> Subject: Re: Factors and Terms
> 
> 
/****************   SNIP                               ******************/

> To be precise, there is a distinction between a divisor of a number
> and a factor of a number. The concept of a divisor is best illustrated
> by the example of "perfect numbers". From ancient times the number
> itself is not considered a (proper) divisor, but 1 is, viz. 
> the definition
> of perfect numbers:
> 
> Perfect numbers are those integers which are the sum of their positive
> proper divisors. The first three perfect numbers are
> 6 = 1 + 2 + 3,
> 28 = 1 + 2 + 4 + 7 + 14, and
> 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248.
> 
> A factorization of a number by convention excludes the divisor 1,
> in order to maintain the desirable property that every number larger
> than 1 can be written as a unique product of prime numbers raised
> to integral powers. The factorization of 6 is 6 = 2**1 * 
> 3**1, of 28 is
> 28 = 2**3 * 7**1, of 7 is 7 = 7**!, etc.
> 
> So, by convention, divisors of n include 1 but exclude n, and factors
> of n include n, but exclude 1. Fuzzy math does not make that sharp
> distinction. One *could* qualify the words by talking about "proper"
> divisors and "prime" factors, but that is rarely done.
> 
> 
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