Finding the Factors of a Number

Date: 09/24/2001 at 17:22:11
From: Kourtney Dubbs
Subject: I need to know what factor is

I need to know how to find the factor of a number.

Date: 09/25/2001 at 16:49:05
From: Doctor Ian
Subject: Re: I need to know what factor is

Hi Kourtney,

Every number greater than 1 has at least two factors: 1 and itself.  
For example, 

  2 = 1 * 2

  3 = 1 * 3

  4 = 1 * 4

and so on. That is, two numbers are 'factors' of another number if you 
can multiply them together to get that number. 

Note that 4 has some other factors besides 1 and 4:

  4 = 1 * 4
  4 = 2 * 2

But these are the only factors of 4. A number like 36, on the other 
hand, has _lots_ of factors:

  36 =  1 * 36
     =  2 * 18
     =  3 * 12
     =  4 *  9
     =  6 *  6

Now, how can I _find_ those factors? Well, I start with the ones that 
_have_ to be there, 1 and 36:

  36 =  1 * 36

Next, I try dividing by the next highest number after 1, which is 2.  
2 goes into 36  18 times, so 2 and 18 are a pair of factors:

  36 =  1 * 36
     =  2 * 18

So then I just keep trying more numbers:

  36 =  1 * 36
     =  2 * 18
     =  3 * 12
     =  4 *  9


Now, note that 5 doesn't work, so I leave that out, and go on to 6:

  36 =  1 * 36
     =  2 * 18
     =  3 * 12
     =  4 *  9
     =  6 *  6

Again, 7 doesn't work, and neither does 8. But 9 works:

  36 =  1 * 36
     =  2 * 18
     =  3 * 12
     =  4 *  9
     =  6 *  6
     =  9 *  4

but it would be silly to add that to my table, because it says the 
same thing as an earlier entry:

  36 =  1 * 36
     =  2 * 18
     =  3 * 12
     =  4 *  9  <--+
     =  6 *  6     |  the same factors
     =  9 *  4  <--+

So I know that I'm done. Any numbers larger than 6 will match up with 
something smaller than 6. So the factors of 36 are 1, 36, 2, 18, 3, 
12, 4, 9, and 6. 

Note that if you don't like to think about division, you can find 
factors another way. Imagine that we have 36 cookies, and we'd like to 
arrange them into rectangles. How many different ways can we do that?

  @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@    36 by 1

  @@@@@@@@@@@@@@@@@@
  @@@@@@@@@@@@@@@@@@                      18 by 2

  @@@@@@@@@@@@
  @@@@@@@@@@@@
  @@@@@@@@@@@@                            12 by 3

  @@@@@@@@@
  @@@@@@@@@
  @@@@@@@@@
  @@@@@@@@@                                9 by 4

  @@@@@@
  @@@@@@
  @@@@@@ 
  @@@@@@
  @@@@@@
  @@@@@@                                   6 by 6

Do you see why this is really the same thing we were doing before with 
division? For example, there is no way to arrange 8 rows of cookies to 
get a rectangle with 36 cookies; this is the same as saying that 36 
isn't divisible by 8. 

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

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