LCM 120, HCF 4

Date: 01/24/2002 at 05:22:32
From: WG
Subject: Factorisation

Question: Find two numbers if their LCM is 120 and their HCF is 4. 
(Give three possible answers).

I have tried listing all the factors of 120 (e.g. 6x20, 4x30), but 
none of them fits the description. 1x120 is wrong because 1 is LESS 
than 4. 2x60 is also wrong because 2 is less than 4 also. Same goes 
for 3x40. 4x30 is wrong because 30 cannot be divided by 4... and 
none of the others works either. :( 

Please help. I would really appreciate it.

Date: 01/24/2002 at 09:01:36
From: Doctor Ian
Subject: Re: Factorisation

Hi,

Let's call the two numbers A and B. If the HCF of the two numbers is 
4, then the prime factors must be

  A = 2 * 2 * (the other prime factors of A)
 
  B = 2 * 2 * (the other prime factors of B)

And the sets of other prime factors can't have any factors in common, 
because then the HCF would be something other than 4.  Does that make 
sense? 

So we're looking for two numbers, both multiples of 4, that share no 
prime factors other than a pair of 2's, and that both divide 120 
evenly. 

How can we find them?  Well, let's look at the prime factors of 120:

  120 = 2 * 60
      = 2 * 2 * 30
      = 2 * 2 * 2 * 15
      = 2 * 2 * 2 * 3 * 5

Suppose I divide up the factors this way:

  A = 2 * 2 * 2 
    = 8

  B = 2 * 2 * 3 * 5
    = 60

What's the largest number that divides both of these numbers?  Looking 
at the prime factors, we can see that it's 4. So the HCF is 4. What's 
the smallest number that both of these will divide evenly?  It would 
be 

      2 * 2 * 2
      2 * 2     * 3 * 5
      -----------------
      2 * 2 * 2 * 3 * 5 = 120
      \___/
        |
        +-- These appear in both numbers, so we only count them once.

So 4 and 60 would appear to be one answer to the problem. Can you find 
some other solutions? 

Note that your assumption that the LCM of two numbers is the product 
of the numbers is sometimes correct, but sometimes not. By thinking 
about prime factors, can you come up with a rule for predicting when 

  LCM(a,b) = a*b

will be true? 

I hope this helps.  Write back if you'd like to talk more about this, 
or anything else. 

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

Date: 01/24/2002 at 09:39:34
From: WG
Subject: Factorisation

Thank you so much :) I never did dream that you would reply so soon! 
I have an ambition to be a mathematician or cosmologist when I 'grow 
up'. If I do become one I'll remember (try to anyway) your homework 
help service :) Thanks again!