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Least Common Multiple

Date: 01/24/98 at 14:09:15
From: kristi
Subject: Least common denominator

I need to find the LCD of the numbers 16,20,9,and 8. I've gone up to 
1,000 and still can't find it. If you could help, please do,


Date: 01/25/98 at 09:52:42
From: Doctor Scott
Subject: Re: Least common denominator

Hi Kristi!

Remember that the LCD is the smallest number that is evenly divisible 
by all of the denominators of the given fractions. Since you were 
given numbers not fractions, technically you are looking for the LCM, 
the LEAST COMMON MULTIPLE, of the numbers.  

Most textbooks recommend that you list the multiples of the numbers 
and then look for the smallest one that is in all of your lists.  
I think, though, that you can start by listing the multiples of the 
largest number, and then ask yourself whether the other numbers "go 
into" the multiples. Let's look at your example:

The multiples of 20 are:  

  20, 40, 60, 80, 100, 120, 140, 160, 180, 200, ...

starting at 20, ask yourself, "does 16 go into 20, evenly?" Since the 
answer is "no," 20 is not the LCM.

now 40, "does 16 go into 40 evenly?" No.
now 60, "does 16 go into 60 evenly?" No.

You might also use what you know about the other numbers to help.  
Since we want 9 to be a factor of our LCM, and we know that all of the 
multiples of 9 have a special property (the sum of the digits is a 
multiple of 9), we can eliminate 80, 100, 120, 140, and 160.

So, let's try 180: 20 is a factor (20x9), 9 is also a factor. Is 16?  

Using the properties of multiples of 9 eliminates 200, 210, 220, 240, 
260, 280, 300, 320, 340.

So let's try 360: 20 is a factor (20x18), 9 is a factor (9x40), 16?  

Properties of multiples of 9 eliminate 380, 400, 420, 440, 460, 480, 
500, 520.

So let's try 540: 9 and 20 are factors, but 16 is not.

We eliminate 560, 580, 600, 620, 640, 660, 680, 700.

Try 720: 9 and 20 are factors, so is 16 (16x45), so is 8 (8x90). Wow!  
We found it.

In general, we would have to test a lot more of the multiples to see 
if the other numbers are factors, but when we are using numbers like 9 
that have some special properties, it makes our work much easier!

-Doctor Scott,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 01/25/98 at 09:58:01
Subject: Re: Least common denominator

Thank you so much! I've been trying to figure this out all weekend and 
now I can finally finish my last problem for my homework.

:-) Kristi
Associated Topics:
Middle School Division

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