Prime Numbers vs. Prime Factors

Date: 01/13/97 at 20:08:07
From: Julianne Stemp
Subject: Prime factor

What is the difference between a prime number and a prime factor?  
What is the prime factor of 18?

Date: 01/15/97 at 18:44:28
From: Doctor Mike
Subject: Re: Prime factor

Hello Julianne,
  
These are good questions.  I'm glad you want to use the words 
correctly.  I'll start out with a word you didn't ask about.
  
A "composite" number is a number like 18 which can be written as 
smaller numbers multiplied together, like 6*3 or 2*9.

A "prime" number is a number like 17 which cannot be written as 
smaller numbers multiplied together without using "1" as one of only 
two numbers.  That is, 17 = 1*17 = 17*1 but those are the only ways 
you can multiply two numbers together to get 17.  Prime is the 
opposite of composite.
   
The numbers we are talking about here are the positive "whole" or
"natural" or "counting" numbers 1,2,3,4,5,6,7,8,9,10,11 and so on.  
The ones of these that are prime are 2,3,5,7,11 and so on.  
   
Let's go back to the example 18 = 6*3 from above.  Any numbers you can 
multiply together to get 18 are called factors of 18.  So 6 is a 
factor of 18 and so is 3.  Because 18 = 2*9 also, it is true that 2 is 
a factor of 18 and 9 is a factor of 18.  Since 18 = 1*18 = 18*1 are 
also true, 1 and 18 are usually called factors of 18 too.  
  
A "prime factor" of a number is a factor of the number which also 
happens to be a prime number.  So, 2 and 3 are prime factors of 18 but 
the others are not.  For instance, 6 is a factor of 18 but it is not a 
prime factor of 18 because 6 is a composite number.  I should add that 
there is not just one prime factor of 18, because there are two of 
them.  
  
There is a special factorization of 18, namely, 18 = 2*3*3.  What 
makes this special is that there are only prime numbers on the right
side of the equal sign.  There is a fact about arithmetic that any
composite number can be written as a product of prime numbers in ONLY 
ONE WAY (if you don't count order).  This special way of writing the 
number 18 is called its prime factorization.  To find it, I could have 
started with 18 = 2*9 and then factored 9 to get 18 = 2*3*3, OR I 
could have started with 18 = 3*6 and then factored 6 to get 
18 = 3*2*3, which is the same as 18 = 2*3*3.  
  
By the way, since we are talking about what words mean, the fact about 
numbers which says that order of multiplication does not make any 
difference (a*b = b*a) is called the "commutative" property. 
  
I hope this gives the information you needed.
   
-Doctor Mike,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/