Prime, Composite, or Neither?

Date: 09/16/2003 at 19:02:12
From: Hillary
Subject: prime and composite numbers

What are prime and composite numbers?  I just don't get it.

Date: 09/17/2003 at 11:01:40
From: Doctor Ian
Subject: Re: prime and composite numbers

Hi Hillary,

Suppose I have 12 items, and I try to arrange them into a rectangle. 
I can do this in more than one way:

  . . . . . . . . . . . .       1 x 12

  . . . . . .                   2 x 6
  . . . . . .

  . . . .                       3 x 4
  . . . .
  . . . .

For some numbers, there is only one rectangle that I can make.  For
example, if I have 7 items, I can do this:

  . . . . . . .

But if I try to make more rows, I always have something left over:

  . . . .
  . . .


  . . .
  . . .
  .

  
  . .
  . .
  . .
  .

A number like 7 is called 'prime'.  In contrast, a number like 12 is
called 'composite'.  

One way to remember this is that something that is 'composed' is 'put
together' from smaller pieces.  (For example, we compose a poem from
words, and compose a song from notes.) 

In the case of a number like 12, we can put it together in more than
one way, using multiplication:

  12 = 1 x 12 

     = 2 x 6

     = 3 x 4

But 1 x 12 is hardly like putting something together, is it?  So if we
ignore ways that include a 1, we see that there are two ways to put
together a 12, 

  12 = 2 x 6

     = 3 x 4

and _no_ ways to 'put together' a 7.  

One tricky point is that the number 1 is considered to be neither
prime nor composite.  (Think about why this would be the case.)  So
while it's tempting to say things like 

   A number is prime if it's not composite.

or

   A number is composite if it's not prime.

neither of these is quite true, because 1 isn't composite, but it's
also not prime; and 1 isn't prime, but it's also not composite. 

Why do we care about any of this?  That's discussed here:

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

Does this help?  Write back if you'd like to talk more about this, or
anything else. 

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