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Finding Factor Pairs for Whole Numbers

Date: 11/29/2005 at 16:48:09
From: Tansu
Subject: What is a factor pair

I know you have already answered this question, but I still need help.  
What are the factor pairs of 66?  Every time I try I don't really 
understand.



Date: 11/29/2005 at 17:12:15
From: Doctor Ian
Subject: Re: What is a factor pair

Hi Tansu,

Let's look at a smaller number, like 18.  If we want to find all the
factor pairs, we can try dividing 18 by every number up to 18, and see
which ones divide it evenly:

  18 /  1 = 18
  18 /  2 =  9
  18 /  3 =  6
  18 /  4 = 
  18 /  5 = 
  18 /  6 =  3
  18 /  7 = 
  18 /  8 = 
  18 /  9 =  2
  18 / 10 = 
  18 / 11 = 
  18 / 12 = 
  18 / 13 = 
  18 / 14 = 
  18 / 15 = 
  18 / 16 = 
  18 / 17 = 
  18 / 18 =  1 

Now, we can notice a couple of things.  The first is that it's
pointless to try any numbers greater than half of 18, because there's
no way they can divide evenly.  (Do you see why?)  Also, a number is
always divisible by itself.  So we could have done this:

  18 /  1 = 18
  18 /  2 =  9
  18 /  3 =  6
  18 /  4 = 
  18 /  5 = 
  18 /  6 =  3
  18 /  7 = 
  18 /  8 = 
  18 /  9 =  2
  18 / 18 =  1

The second thing to notice is that each of these divisions corresponds
to a multiplication.  That is, 

  18 /  1 = 18    ->   18 = 18 x 1
  18 /  2 =  9         18 =  9 x 2
  18 /  3 =  6         18 =  6 x 3
  18 /  4 = 
  18 /  5 = 
  18 /  6 =  3         18 =  3 x 6
  18 /  7 = 
  18 /  8 = 
  18 /  9 =  2         18 =  2 x 9
  18 / 18 =  1         18 =  1 x 18

So these are the factor pairs of 18:  

  18 and  1
   9 and  2
   6 and  3
   3 and  6
   2 and  9
   1 and 18

Does this make sense?  At this point, a second thing we can notice is
that the later part of the list is the same as the earlier part, but
with the factors in the opposite order:

  18 and  1 --------
   9 and  2 -----   |
   6 and  3 --   |  |
              |  |  |  These are the same, with order reversed
   3 and  6 --   |  |
   2 and  9 -----   |
   1 and 18 --------

So really, we can stop dividing when the quotient is smaller than the
number we're dividing by:

  18 /  1 = 18
  18 /  2 =  9
  18 /  3 =  6
  18 /  4 =  4 remainder 2
  18 /  5 =  3 remainder 3    <--  We can quit here

There's no reason to continue after this point, because any pair we
would get, we'll already have found.  

In general, the stopping point is reached when we get to the square
root of the number whose factor pairs we're finding.  For 18, the
square is somewhere between 4 (4^2 = 16) and 5 (5^2 = 25), so 5 is the
upper bound of the numbers we want to check. 

For 66, the square would be between 8 (8^2 = 64) and 9 (9^2 = 81).  So
although 66 seems like a much bigger number than 18, there aren't a
lot more numbers to check:

  66 / 1 =  
  66 / 2 = 
  66 / 3 = 
  66 / 4 = 
  66 / 5 = 
  66 / 6 = 
  66 / 7 = 
  66 / 8 =
  66 / 9 =

Does this make sense?

Let me know if this helps... or if it doesn't. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/ 



Date: 11/29/2005 at 18:30:50
From: Tansu
Subject: Thank you (What is a factor pair)

Dear Dr. Ian

Thank you so much for replying!  It means a lot to me, it really does.
Everytime I try to write to someone else they NEVER reply.  Now I know
I can rely on you guys.....and understand things when I need help!

Thanks again!

- Tansu (aka Tunz)
Associated Topics:
Middle School Factoring Numbers

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