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### Factoring and Divisibility Rules

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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.
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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/
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Associated Topics:
Middle School Factoring Numbers

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