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### Common Multiples of 4 and 7

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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?
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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/
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Associated Topics:
Elementary Multiplication

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