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### Greatest Common Factors, Least Common Multiples

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Date: 8/29/96 at 20:35:44
From: Anonymous
Subject: Greatest Common Factors, Explained!

Please explain greatest common factors, how to find them, and how they
are different from lowest common multiples.

Thank you,
Amber
```

```
Date: 8/30/96 at 13:41:11
From: Doctor Tom
Subject: Re: Greatest Common Factors, Explained!

Hi Amber,

The "greatest common factor" is also often called the "greatest common
divisor", or GCD, and it is the largest number that divides all of the
numbers in question.

For example, the GCD of 12 and 18 is 6, since 6 divides both numbers
evenly and nothing larger does.

A couple of other examples:

the GCD of 7 and 17 is 1
the GCD of 7 and 49 is 7
the GCD of 100 and 72 is 4

Here's a method to find the GCD of two numbers, which I will show by
example.  Apply exactly the same method in your case. I'll find the
GCD of 12070 and 41820:

41820 = 3*12070 + 5610
12070 = 2*5610 + 850
5610 = 6*850 + 510
850 = 1*510 + 340
510 = 1*340 + 170
340 = 2*170 + 0

As soon as the remainder is 0, you've got the answer: 170.

These are big numbers; usually the number of steps to the GCD is much
smaller.

If you need to find the GCD of three numbers, A, B, and C, for
example, let D = GCD(A, B) and your answer will be the GCD of D and C,
and so on.

The least common multiple is the smallest number that all your numbers
divide, which is sort of the opposite of the GCD.

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

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