Help with Factoring

Date: 4/14/96 at 3:25:24
From: Anonymous
Subject: Pre-algebra prime numbers and factoring

I'm using Multimedia Pro One CD Rom Pre Algebra.  I've just 
started and can't figure out a logical way to get the hang of 
factoring.  I don't want to quit before I get started; but the 
lightbulb hasn't gone on!  Any help?

Date: 4/14/96 at 14:34:59
From: Doctor Syd
Subject: Re: Pre-algebra prime numbers and factoring

Hello!

Are you factoring polynomials or just regular integers?  I assume 
you mean regular integers.  The more you practice factoring, the 
easier it will seem, I promise.  But, to get you started here are 
some ideas:  

First, figure out if 2 divides the number you are factoring.  This 
is an easy step since if the number is even, then 2 must divide 
it, and if not, then 2 doesn't divide it.  Now, if 2 does divide 
the number, figure out what the number divided by 2 is.  Now, take 
this number and figure out whether or not 2 divides it.  Keep 
doing this, keeping track all of the time of all of the 2's you've 
taken out, until you get a number that is not divisible by 2.  
Then repeat the same procedure with 3, 5, 7, etc. (all the prime 
numbers), until you have a product of prime numbers.  This method 
is a method you can always follow that will give you the 
factorization of a number, but it is not always the quickest 
method.  After you have had some practice factoring, you will 
recognize that certain numbers divide certain other numbers, and 
this will make your work much easier.  

Basically, the goal is to write down your given number as a 
product of prime numbers, right?  So, you can just use the 
strategy of writing your number as a product of any two numbers, 
and then writing each of those numbers as a product of two 
numbers, until you reach all prime numbers.  Does that make sense?

You can also use lots of tricks to determine whether or not a 
given number divides another given number.  For instance, there is 
a rule that tells us that if the digits of a number add up to 3, 
then 3 divides that number.  

For instance, suppose we want to factor 108.  Certainly 2 divides 
108 since 108 is even, right?  So, what is 108/2?  Well, 
108/2 = 54.  So, we know that 108 = 54 * 2.  Okay, this means we 
are getting closer, right?  We have written 108 as a product of 
two numbers, one of which is prime.  So, let's write 54 as a 
product of numbers.  If we follow the first strategy described, 
we'd write 54 = 2 * 27.  We'd then write 27 = 3 * 9.  We'd then 
write 9 = 3 * 3.  

Thus we'd get that 108 = 54 * 2 = 2*27*2 = 2*3*9*2 = 2*3*3*3*2, 
thus we've written 108 as a product of primes.  Does that make 
sense?  

I hope this helps.  Write back if you have more questions.

-Doctor Syd,  The Math Forum