



Most
numbers out there in numberland 


are
made up of two or more smaller numbers multiplied together. 


Like
say, the number 12. 


If
we think a bit we can think of two numbers 


that
we can multiply together to get 12. 





Like
maybe 4 x 3?






4
x 3 = 12






It's
like the 4 and the 3 were hiding inside the 12 


and
they just popped out. 


We
can draw that like this: 











Swell,
now we have a 4 and a 3. 


Could
we break up the 4 or the 3 like we broke up the 12? 


Well
2 x 2 = 4. 


That
means we can break up the 4. 











OK,
how about the 3? 





3
x 1 = 3






Well
yeah, but: 





Any
number x 1 = That number 





In
fact:






Any
number x 1 x 1 x 1 x 1 x 1 x 1 x 1 = That number






So
we really don't get too excited about the "1 times the
number" deal. 


Great,
are there any other whole numbers that we can multiply together to
get 3? 


I
don't think so. 


That
means there's nothing that we can do with the 3. 


What
about those 2's? 


Are
there any numbers that we can multiply together to get 2? 


Only
the 2 times 1 thing and we already said we don't like that. 


If
we can't break up the numbers any more, we're done. 





Numbers
that we can't break up any more are called PRIME
NUMBERS. 


All
the numbers at the bottom of the paths in the picture are prime
numbers. 


If
we collect them all and multiply them together, 


we
get the number at the top. 


(Unless
we made a mistake that is.) 











2
x 2 x 3 = 12






Remember,
since we can't break up the 2's and 3 any more 


they
are called prime numbers. 


AND,
since 2 x 2 x 3 = 12: 





2,
2, and 3 are the PRIME FACTORS of 12






Let's
try another one ... 





Example: 





Break
up 24 into prime factors. 


OK,
24 is an even number so it has to be equal to 2 times something. 


But
2 times what? 


Whatever
it is will be half as big as 24. 


2
x 10 = 20 so it's bigger than 10. 


2
x 11 = 22 close. 


2
x 12 = 24 BINGO! 








We
know from the last problem that 2 is a prime number 


so
that side of the game is done. 


What
about the 12? 





Well
12 is an even number so it is equal to 2 times something. 





12
= 2 x 6












We
got another 2 so that branch is done, 


what
about the 6? 


6
is also an even number, 


so
it is equal to 2 times something. 





6
= 2 x 3












This
time we got a 2 and a 3. 


The
2 is a prime number, what about the 3? 


It's
a prime number to so that part is finished. 


Now
gather up the numbers at the bottom of the branches. 





Multiply
them together and you get 24 





2
x 2 x 2 x 3 = 24



So
2, 2, 2, and 3 are the Prime Factors of 24 


There
are other numbers besides 2 and 3 


that
can't be broken up into smaller numbers. 


5,
7, and 11 are all prime numbers, and there are a lot more. 


In
fact, there are more than we could ever write down. 





There
is one more trick to all of this. 


What
if you have something like 8 to split up? 


What
do you do with it? 


Here's
the deal. 


We
don't like to use 1 x some number in these 


because
you could put as many 1 times something into any of these. 


But
when we have a negative number, 


we
use: 






1 x (the number)






So
the first step when you have something like  8 is: 






8 =  1 x 8






Now
just leave the  1 alone and work on the 8. 











So: 



1 x 2 x 2 x 2 =  8






copyright 2005 Bruce Kirkpatrick 
