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Prime Factorization

Date: Thu, 26 Jan 1995 15:59:27 AST
From: Richard Seguin
Subject: Prime Factorization

Could one of you guys give me a crash course on Prime 
Factorization.  We just started that Section at school today, 
and since I come from a different province, I know nothing 
about it.

In class they can divide so quickly, is there a trick to 
dividing 2 numbers so quickly?

Date: 26 Jan 1995 15:59:31 -0500
From: Elizabeth Weber
Subject: Re: Prime Factorization

Hello there!

Prime factorization is pretty simple once you know what's going
on, and it's fun too!  I used to do it in the margins of my notebooks 
when I was bored in class.  

Every whole number can be put into one of two categories:  prime
or composite.  Prime numbers are numbers that you can't divide 
without getting a fraction, unless you feel like dividing them by 
themselves or by 1.  These are numbers like 3, or 5, or 7, or 19, or 
379721 (I told you I used to do this when I was bored!).  Composite 
numbers, on the other hand, can be split up into something 
smaller;  8 is 2 times 4, 4 is 2 times 2, 21 is 3 times 7, 100 is 5 times 
5 times 2 times 2, and so on.  Thus, 8, 4, 21, and 100 are all 
composite numbers.  It's easy to remember which word means
which--composite has the word compose in it, and you can make 
composite numbers out of smaller numbers just like you can 
compose a piece of music out of smaller sounds.  
Now, the number 1 is a problem.  I've heard people say that it's
prime, since you can divide it by 1 without getting a fraction.  I've
also heard people say that it has its own little category, because a 
prime number should be divisible by two numbers: 1 AND itself, 
and nothing else; and since 1 is one, you can only divide it by 1 
number without getting a fraction.  I'd ask your teacher whether
she thinks it's prime or not, but it doesn't really matter, at least 
not when you're doing prime factorization.  

All prime factorization is, is taking a composite number and splitting
it up into the little numbers that it's made up of until you can't split
it up any more.  Take the number 8, for example.  
We can divide 8 into 2 and 4.

                        / \
                       2   4   But we're not done yet;  4 can be split 
                                                        into 2 and 2
                           / \  
                          2   2    Now we're done.

If you draw the smaller pieces (called the factors) in an upside down 
tree, like I just did, you can go back and collect all the pieces at the 
ends of the branches, and they will be the "prime factors" of the 
number.  So, the prime factors of 8 are 2, 2, and 2.  

Here's another example:  30

         /  \
        5    6
            / \
           2   3

So, the prime factors of 30 are 2, 3, and 5.

Now, you wanted to know some tricks of dividing:

2--every even number is divisible by 2;  if a number ends in a 2, 4, 6, 8,
or 0, at least one of its factors will be a 2.

3--if you add the digits of a number, and the number you get is divisible
by 3, then the original number is divisible by 3.  For instance, if you
take the number 57, and you add the digits, 5+7=12, and since 12 is
divisible by 3, 57 is divisible by 3;  3 is a factor of 57.

4--take the last 2 digits of the number.  If they are divisible by 4, then
the number is divisible by 4.  216, for example, is divisible by 4, because
16 is divisible by 4.

5--anything that ends in a 5 or a 0 is divisible by 5.  

6--anything that is divisible by 2 and by 3 is divisible by 6.  Can you
figure out why?

7--Somebody once told me that there was a shortcut to finding out if a
number was divisible by 7, but she said that it would take just as long to
walk to the store and buy a calculator as to use it, so it really wasn't
much of a short-cut.

8--If the last 3 digits of the number are divisible by 8, the number is
divisible by 8.

9--If the sum of the digits of the number is divisible by 9, then the
number is divisible by nine.

10--If a number ends with a 0, it's divisible by 10.

12--If a number is divisible by 3 and by 4, it's divisible by 12.  Can you
figure out why?

Those are all of the short-cuts to finding factors that I know.  If you
have any more questions, or if any of this doesn't make sense to you, 
write back to us!

For summaries, see "Divisibility Rules" and "Explaining 3, 9, 11, 7, 13, 17, 
and larger numbers":   

-Dr. Elizabeth  The Math Forum
 Check out our Web site!    

Associated Topics:
Middle School Factoring Numbers

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