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Common Multiples of 4 and 7


Date: 11/19/97 at 23:53:14
From: Leng leng
Subject: Common multiples of 4 & 7

I wonder what are the numbers that are common multiples of 4 and 7.

I used the 4 first - 4,8,12,16, etc. - and the 7 is 7,21,28, etc. 
Am I right?


Date: 11/25/97 at 16:25:46
From: Doctor Mark
Subject: Re: Common multiples of 4 & 7

Hi!

You got the multiples of 4 all right, but you missed one of the
multiples of 7 (that ends up not making a difference in the answer, 
though). 

This is easy to do, so let's see how to do it systematically.

Remember that a multiple of 4, say, is just 4 times another integer.  
Said another way, if you were to count by 4's, starting at 4, you 
would automatically list all the (positive) numbers that are multiples 
of 4:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, and so on.

Similarly, you can list all the (positive) multiples of 7 by starting 
at 7, then counting by 7's:

7, 14, 21, 28, 35, 42, 49, 56, 63, and so on.

Now if we are asked to find the *common* multiples of 4 and 7, then 
the word "common" tells us that we want to find a number which the 
lists of multiples of 4 and of 7, have in "common" - we want to find a 
number that occurs in both of our lists. If you look closely, you will 
see that 28 appears in both lists, as does 56.

Why don't you continue both lists of the multiples of 4 and 7 (which 
is pretty easy to do if you use a calculator), and see what the other 
common multiples of 4 and 7 are?  Once you have done that, make a list 
of the common multiples you just found, and then next to it, write 
down the multiples of 28.  What do you notice?  Do you see how to find 
all the common multiples of 4 and 7 without writing down all the 
multiples of 4 and the multiples of 7?

When you get further along in school, you will learn another way of 
finding the common multiples of any two numbers, using something 
called the Prime Factorization. Once you learn that, then you will 
find it easy to find the common multiples without having to write out 
a long list.

One other thing. Although it turned out in this case that the smallest
common multiple of 4 and 7 was just 4 times 7 (which is, of course, 
28), it's not always true that you find the smallest common multiple 
by multiplying the two numbers. To convince yourself of that, try 
finding a list of the common multiples of 4 and 6. What's the smallest 
common multiple?  Is it 4 times 6?

Good luck, and write back if you have any other questions.

-Doctor Mark,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
Elementary Multiplication

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