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Product Equal to 12

Date: 04/09/2003 at 17:44:21
From: Jan W. Pollard
Subject: How many different numbers have a product equal to 12? 

How many different numbers have a product equal to 12? 

I need an elementary example.  What type of math is this called?


Date: 04/10/2003 at 15:50:43
From: Doctor Ian
Subject: Re: How many different numbers have a product equal to 12? 

Hi Jan,

This is called 'factoring'. If the product of two numbers is another
number, then the first two are factors of the third. 

For example, 

   Product   Factors
   -------   -------
     12      1, 12         because 1*12 = 12
     12      2, 6                  2*6  = 12
     12      3, 4                  3*4  = 12

So the factors of 12 are 

  1, 2, 3, 4, 6, 12

The simplest way to find the factors of any number is to start 
dividing it by every smaller number. If you get no remainder, it's a
factor. For example, 

  12/1  = 12 (no remainder)         So 1 is a factor
  12/2  =  6 (no remainder)         So 2 is a factor
  12/3  =  4 (no remainder)         So 3 is a factor
  12/4  =  3 (no remainder)         So 4 is a factor
  12/5  =  2 (remainder 2)
  12/6  =  2 (no remainder)         So 6 is a factor
  12/7  =  1 (remainder 5)  
  12/8  =  1 (remainder 4)  
  12/9  =  1 (remainder 3)  
  12/10 =  1 (remainder 2)  
  12/11 =  1 (remainder 1)  
  12/12 =  1 (no remainder)         So 12 is a factor

Now, if you do this for a few numbers, you should quickly see some
patterns. For example, once you get up to half the product, you never
find any more factors until you get to the product itself. Can you
see why that has to be true? 

So you can stop dividing when you get halfway to the product. 

A second pattern is this: factors come in pairs. So in the example
above, when I divided by 3 and got 4, with no remainder, that told me
that I didn't really need to divide by 4. Why? Because I'd get 3 with 
no remainder. Similarly for 2 and 6.  

Of course, for a big number like 100, that's a lot of dividing. And
as you might expect, mathematicians have figured out ways to eliminate
most of the work, by using 'prime factors'.  You can read about how to
do that here:

   Finding All the Factors of 100
   http://mathforum.org/library/drmath/view/58525.html 

and here:

   Why Study Prime and Composite Numbers? 
   http://mathforum.org/library/drmath/view/57182.html 

and

   Prime Factoring
   http://mathforum.com/library/drmath/sets/shortcuts/
dm_prime_factors.html

Does this help? 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Factoring Numbers

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