Prime Factorization

Date: 01/22/97 at 20:58:13
From: Jayme
Subject: Prime Factorization

The other day while I was gone, my class learned about prime factors. 
Now we have a homework sheet that says "Write the prime factorization 
for each number". I don't know what it means. Could you please help 
me?

Date: 01/22/97 at 21:37:18
From: Doctor Wallace
Subject: Re: Prime Factorization

Hi Jayme!

Let's begin with some definitions.

A factor is an integer that exactly divides a given integer.

For example, if we are given the number 12, we can list its factors as 
the following:  1, 2, 3, 4, 6, and 12.  This is because each of these 
numbers goes into 12 evenly, with no remainder.  Notice that 5, for 
example, is not listed, because 12 divided by 5 leaves a remainder.

Now, a prime number is one that has only 2 factors: 1 and itself.

For example, 7 is a prime number, because if we list out all its 
factors, we only have 1 and 7 on the list.  No other number divides 
into 7 without leaving a remainder.

Now, we can combine these two ideas and ask about the prime factors of 
a number.  For example, what are the prime factors of 12?  Answer: 2 
and 3. (We can also include 1, but since it is a factor of every 
number, it is usually omitted from such lists.)  These are the only 
two prime numbers in the list of the factors of 12, so they are 12's 
prime factors.

Now for your homework.  It asks you to write the "prime factorization" 
of a number.  That's just a little different from writing a number's 
prime factors.  It's important that you understand this.  We listed 2 
and 3 as the prime factors of 12, but they are NOT the prime 
factorization of 12.  The two ideas are different.

So what is prime factorization?  Well, there is a theorem in 
mathematics that says that any integer can be expressed as the product 
of its prime factors.  This means that we can write a number using 
only the product of its prime factors, and we are allowed to repeat 
some of them if we need to. 

Let's look at an example.  Let's take our friend 12.  We've already 
seen that the prime factors of 12 are 2 and 3.  So how can we write 12 
as a product of these factors?  Well, 2 x 3 = 6.  If we multiply 6 by 
2, we'll get 12.  So 12 can be written as 2 x 2 x 3.  This is called 
the prime factorization of 12.  (The numbers could be written in any 
order, but usually we write the smaller ones first, and then go to the 
bigger.)

Let's do another one.  How about 10?  The prime factors of 10 are 2 
and 5. Since 2 x 5 = 10, the prime factorization of 10 is 2 x 5.

I don't know what kinds of numbers are on your homework sheet.  If 
they are small numbers, like these in my examples, you should be able 
to figure them out without too much trouble.  Just remember that in 
the prime factorization, you can only use prime numbers, and they must 
be factors of the number you're trying to find the prime factorization 
for.  You may use any of the number's prime factors as many times as 
you like.  And you know what else?  Each number has only 1 prime 
factorization.  Finding it is a little like a puzzle.

Suppose you have larger numbers.  For example, how would you find the 
prime factorization of 126?  Well, one way you could start would be by 
noticing that 126 is even.  2 is the only even prime number, and it 
divides evenly into every even number.  So, if we divide 126 by 2 we 
get 63.  We know from our multiplication tables that 63 = 9 x 7.  We 
know that 7 is prime; what about 9?  Nine is not prime: 9 = 3 x 3.  So 
now we can stop since we have reached only prime numbers.  So the 
prime factorization of 126 is:

           126 = 2 x 3 x 3 x 7      
               = 2 x   9   x 7
               = 2 x    63  
               = 126

I hope that makes it easier to see.  If you start with an odd number, 
you can guess what might go into it, until you find a factor.  Try 
prime numbers, of course, like 3, 5, 7, 11, and 13.  When you find one 
that works, then divide it into your number, and repeat the process 
with the new, smaller number you get, just as we did above.

I hope this helps, Jayme.  If you need more help, don't hesitate to 
write back!

-Doctor Wallace,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 01/26/97 at 16:52:37
From: Doctor Mitteldorf
Subject: Re: Prime Factorization

Dear Jayme,

When you factor a number, you're just finding some numbers that can be 
multiplied together to make that number.  You can factor 24 by saying 
6*4 or you could just as well say 12*2 or 8*3.

Prime factorization is when you finish the process by breaking down 
each of your factor numbers, and keep going until you get to numbers 
that aren't divisible by any others (except, of course, 1 or the 
number itself).

For example, if you said 24 was 6*4, you might change the 6 to 3*2 and 
the 4 to 2*2, and you'd have 24 = 3*2*2*2.

If you said 24 was 8*3, you'd break down the 8 into 2*2*2, and you'd 
end up with the same thing as before: 24 = 3*2*2*2.

(The order of the numbers doesn't matter.)

Hope this helps.  Write again if you have more questions about prime 
factors.

-Doctor Mitteldorf,  The Math Forum
 Check out our web site!