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### Precise Numbers

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Date: 10/18/2001 at 21:28:55
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

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
Subject: Re: Precise Numbers

Yes - thank you very much.

Ryan
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