Finding All the Factors of a Number

Date: 09/10/2001 at 22:14:40
From: Shaya
Subject: I need help with factors

What are the factors of 1-100? Like 1 = 1x1, and so on.

Date: 09/11/2001 at 10:48:54
From: Doctor Ian
Subject: Re: I need help with factors

Hi Shaya,

Do you know what a prime number is? A number is prime if it cannot be 
divided evenly by anything except itself and 1. For example, 5 is a 
prime number, because the only factors of 5 are
 
  1 * 5 = 5 

However, 12 is _not_ a prime number, because   

  1 * 12 = 12
  2 *  6 = 12
  3 *  4 = 12 

Why should you care about prime numbers? For one thing, they make it 
easy to find the factors of a number. 

Let's consider a number like 84. It's even, so it's divisible by 2:   

  84 = 2 * 42 

Now, what about 42? That's also even, so it's divisible by 2:   

  84 = 2 * 2 * 21 

What about 21? That's not divisible by 2, but it _is_ divisible by 3: 

  84 = 2 * 2 * 3 * 7 

And 7 is prime. So what we've done here is find the 'prime factors' of 
the number 84. 

How does this help us find _all_ the factors of 84? Well, if a number 
is a factor of 84, and it's not one of these prime factors, then it 
must be a product of two or more prime factors. So we can find all the 
factors of 84 by looking at all the ways we can group these prime 
factors:   

  2 * 2 = 4        pairs
  2 * 3 = 6
  2 * 7 = 14
  3 * 7 = 21   

  2 * 2 * 3 = 12   triplets
  2 * 2 * 7 = 28
  
  2 * 3 * 7 = 42 

So the factors of 84 are   

  1                           A factor of everything.

  2, 3, 7                     The distinct prime factors.

  4, 6, 14, 21                Products of two prime factors.

  12, 28, 42                  Products of three prime factors.

  84                          Product of all the prime factors. 

This was a tough one! Most numbers won't have so many prime factors.  
For example, consider 35.  35 = 5 * 7, so the prime factors of 35 are 
5 and 7:

  1                           A factor of everything.

  5, 7                        The distinct prime factors.

  35                          Product of all the prime factors. 

I know you'd probably like me to just give you a list of all the 
factors of all the numbers from 1 to 100, but the point of this 
exercise is for you to get some practice manipulating numbers. It may 
seem boring, but the effort you put in now will make things much, much 
easier for you later. 

I hope this helps.  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/