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One Approach to Finding LCM and GCF at the Same Time

Date: 03/02/2006 at 22:00:24
From: Lindsay
Subject: This is a LCM method helper 

I just wanted to show you something I learned recently on how to find
the LCM and GCF at the same time.  You make a chart with the two
numbers in question, let's say 12 and 8:

      12 ** 8

Now you decide what number can go into both, like 2 or 4.  Let's
choose 4, and write it on the left:

      12 ** 8
  4*

Now ask how many times does 4 go into 12 and put it under the 12, 
then same with 8:

      12 ** 8
  4*   3 ** 2

Now make sure you can't put any number other than 1 into 3 and 2 (if 
we chose 2 in the first place we would have to do the process again).
Anyway we're done with this one, now the 4 in the left is the GCF 
and then we multiply the GCF with the bottom row (in this case the 3 
and 2) ... 4 x 3 x 2 = 24 which is our LCM.

This also works with 3 numbers but they have to all have a GCF other 
than 1 otherwise it's more work and not really worth the hassle.  Hope
this can help others!



Date: 03/02/2006 at 23:50:34
From: Doctor Peterson
Subject: Re: This is a LCM method helper

Hi, Lindsay.

Thanks for sharing your method.  I like it!  It corresponds to what I
recommend in simplifying fractions, that you take it in steps, just
dividing by whatever you see that is a common factor, and then repeat
until there's nothing left.  In fact, that's exactly what you're doing
here, with the bonus of getting the LCM at the end.  Let me try it in
a more complicated case.

Suppose we want to find the GCF and/or LCM of 54 and 90.  I first see
that they are both even, so I divide by 2:

    | 54 | 90
  --+----+----
  2 | 27 | 45

Now I see that 27 and 45 are both multiples of 9, so I divide by that:

    | 54 | 90
  --+----+----
  2 | 27 | 45
  9 |  3 |  5

Now I can't divide any more.  The GCF is the product of the numbers
down the side, 2*9 = 18, and the LCM is the product of the whole L on
the left and bottom, 2*9*3*5 = 18*15 = 270.  At the same time, the
ratio 54:90 or the fraction 54/90, simplified, is sitting there as
well, 3:5 or 3/5.

Also, it seems very natural that the product of the factors should be
the GCF, so it's easy to remember how to get that.  The LCM is more
magical, but I can see how to explain why it works: in order to get a
multiple of 54, we have to multiply the GCF by the 3, and in order to
get a multiple of 90 we have to multiply by the 5, so using both gives
us a common multiple.

How about three numbers?  I don't think the LCM part would work at
all.  I'll try this one:

    | 54 | 90 | 70
  --+----+----+----
  2 | 27 | 45 | 35

No, the LCM isn't going to work, since we only need one extra 5, for
example, not both 45 and 35 in the mix--do you have a way to do it?  
I think I can come up with something, but not just now.

I usually use the method based on prime factors, particularly because
it extends naturally to factoring monomials; but I think this might
work well there too.  You can start right off dividing out the 
variables, and then take out the numerical factors as well:

    | 54xy^2 | 90xy
  --+--------+------
  x | 54y^2  | 90y
  y | 54y    | 90
  2 | 27y    | 45
  9 |  3y    |  5

So the GCF is 18xy and the LCM is that times 15y, or 270xy^2.  Not
bad.  The only thing missing is getting the LCM easily for more than
two numbers, which the prime factor method handles easily.

Do you have a reference for the source of this idea?  And is what I've
done more or less what you learned, or an extension of it?


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
Middle School Factoring Numbers

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