Pre Algebra Prime Factorization
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The Big Break-up
Prime Factorization

 
 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

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