Factoring and Divisibility Rules

Date: 12/18/97 at 19:41:10
From: Kate
Subject: Factoring 

I have no idea how to factor these problems:

12*13-60+12(squared)

11(squared)-6*11+5*11

I'm not sure where to start.

Date: 01/10/98 at 10:52:02
From: Doctor Marko
Subject: Re: Factoring 

Kate,

The word factoring is used in two contexts:

1. Factoring a number into its prime factors (like for example 
   12 = 2*2*3)

2. Factoring the expression such as the ones you have above so that 
   you simplify the expression so you can do calculations with it more 
   easily. 

In order to learn this one needs to know how to factor a number into 
its prime factors (see no. 1) and so I will explain that as well, even 
if it is only a review.

If your question concerns factoring in the sense of no. 1, the recipe 
is to know the 'rules' of divisibility:

    i) even numbers are divisible by 2.
   ii) if the sum of the number's digits is divisible by 3 then that 
       number is divisible by 3.
  iii) if the last two digits of a number are divisible by 4 then the 
       number is divisible by 4.
   iv) if the number ends in 5 or 0 it is divisible by 5.

Let's try number 105. You may be able to automatically say that it is 
divisible by 5, but it is customary to start from 2 and work your way 
up. So 2?  No, the number is odd. 3?  Well, 1+0+5 = 6, which is 
divisible by 3, and so 105 is divisible by 3. 105/3 = 35.  (REMEMBER 
3!)

Now we have factored 105 into 3 and 35, but 35 is not a prime number, 
so we start the whole factoring procedure again, only now for 35.

Is 35 divisible by 2? No, because it is odd. How about by 3? No, it is 
not, since the sum of its digits (3+5 = 8) is not divisible by 3. It 
is not divisible by 4 either, since it is not divisible by 2, but it 
is divisible by 5, since 35 ends in a 5. 35/5 = 7.  (REMEMBER 5!)

7 is a prime number also, so we are done - we have the prime 
factorization of 105, which is 3*5*7. Good. It becomes harder when the 
prime factorization has larger prime numbers like 19 and 37, but in 
that case I am not sure that there is a safe recipe other than luck.

However, it sounds like you are more interested in the no. 2 
connotation of the word factorization.  For that, follow these steps:

a) Determine the biggest common factor among your numbers. Here the 
   emphasis need not be on THE BIGGEST, but the bigger it is the 
   easier it will be for you to factor.

   -- So it is clear that in your second example it must be 11, since 
   every number in the sum is written as 11 times something. In the 
   first case it is a bit tougher. The first thing you need to do is 
   figure out the prime factorization of 60 by the method in no. 1.  
   The factorization is 2*2*3*5, which you can write as 12*5, as well.  
   So, now you have the same situation as in the second example, and 
   clearly your biggest common factor is 12.

b) Figure out the remaining factors of each of your numbers:
   -- In the first case those are 13, 5, and 12.  And in case two they 
   are 11, 6, and 5.

c) Now you are ready to factor:
   -- First write your common factor which will be multiplied by a 
   bracket.  

Inside the bracket you will have the remaining factors in the correct 
order and with the right signs. Then add the remaining factors and you 
are done.

To make it more concrete let me completely work out one of your 
examples:

   12*13 - 60 + 12*12  Common factor is 12, so
   12*(13 - 5 + 12) = 12*20 = 240

Here you can see the power of factorization, since we solved that 
messy problem without doing much multiplication. 

Hope this has helped.  If you need some additional help, please do not 
hesitate to write.

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