Precise Numbers

Date: 10/18/2001 at 21:28:55
From: Ryan Adams
Subject: Precise Numbers

I have a contest problem about precise numbers which the problem 
alleges is in the same category as proper divisors. I can't find any 
formulas to figure them out.  

The problem says that a precise number occurs when its proper divisors 
multiplied together equal the number. For example, 6 is a precise 
number because 1*2*3 = 6. I'm wondering if there is another name or 
formula that would help me figure out the problem.

Thanks,
Ryan

Date: 10/19/2001 at 17:32:26
From: Doctor Ian
Subject: Re: Precise Numbers

Hi Ryan,

I'm not sure what kind of 'formula' you're looking for, but maybe this 
will be helpful. 

First, just to make sure that you know what it means to break a number 
into prime factors, take a look at 

   Finding All the Factors of a Number
   http://mathforum.org/dr.math/problems/shaya.09.10.01.html   

If a number has exactly two prime factors, p and q, then the only 
proper factors of the number will be 1, p, and q; and when you 
multiply these together, 

  1 * p * q 

you get the number itself. 

Note that if you have more than two prime factors (p, q, and r), then 
you get extra proper factors: 1, p, q, r, pq, pr, qr.  When you 
multiply these together you get something larger than the original 
number:

  1 * p * q * r * pq * pr * qr
  \___________/
    this is 
    the number
    itself

On the other hand, if the number is prime, then the only proper 
divisor is 1.

So in order for a number to be what you're callling a 'precise 
number', it has to have exactly two prime factors. 

So now we can start cranking out 'precise numbers':

   *   2   3   5   7  11
  
   2   4   6  10  14  22
   
   3       9  15  21  33
  
   5          25  35  55

   7              49  77

  11                 121

and so on.  So I guess this is a kind of 'formula' after all.

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 10/20/2001 at 13:02:42
From: Ryan Adams
Subject: Re: Precise Numbers

Yes - thank you very much.

Ryan