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# How to find Greatest Common Factor of whole numbers?

Posted on 2007-07-24
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Please provide procedure for finding Greatest Common Factor(GCM) of whole numbers.  For example, what is the GCM of 27 and 72.
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Question by:naseeam

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In the first website above, I could not find any information on how to find Greatest Common Factor of two whole numbers.

The second website above is too advance for my needs.  I am asking a Fifth Grade Math question.
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Accepted Solution

An easy way is to find the prime factors of both whole numbers. Then find all the divisors that are common for those numbers, and multiply them together - the result is the GCD or greatest common divisor.

Example for 27 and 72 :

27 = 3 * 3 * 3
72 = 2 * 2 * 2 * 3 * 3

The common divisors for 27 and 72 are thus 3 and 3 (twice), so the GCD is 3 * 3 = 9

Here's a more beginner friendly page :

http://en.wikipedia.org/wiki/Gcd

The algorithms that ozo posted are more advanced, and are preferrable for bigger numbers. The technique I illustrated is only feasible for small numbers.
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Assisted Solution

At the 5th grade level; your teacher is likely expecting something like Infinity08 provided.

1) List # = f1 * f2 * ... * fn (for each number)
2) simplify down to simpliest factors

72 ----- 2 * 36
|------- 2 * 18
|----------2 * 9
|-------3 * 3

3) You arrive at Infinity08's approach above (2*2*2*3*3)

4) Do same for other number(s)

5) Multiply highest factor that is common to both lists (in this case 3*3 = 9)

---> 27 / 9 = 3
---> 72 / 9 = 8

There are some other ways involving number layouts... I'll leave this to your teacher to show (as it often took 2 or 3 days for students to grasp when explained in class.

I'd hate to confuse you over the site.
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Assisted Solution

First rule:  The GCD of 0 and any other number x is x.  GCD(0,x) = x
Second rule:  The GCD of any two nonzero numbers x and y is the same as (assuming x >= y) the GCD of (x modulo y) and y.  GCD(x,y) = GCD(x modulo y, y)

So, to solve GCD(27,72):
The second rule says that GCD(27,72) = GCD(72 modulo 27, 27)
72 modulo 27 = 18, so
GCD(27,72) = GCD(18,27)
Using the second rule again, GCD(18,27) = GCD(27 modulo 18, 18)
27 modulo 18 = 9
GCD(27,72) = GCD(18,27) = GCD (9,18) = GCD(18 modulo 9 ,9)
18 modulo 9 = 0
GCD(27,72) = GCD(18,27) = GCD (9,18) = GCD(0,9)

By the first rule, GCD(0,9) is 9.  Therefore GCD(27,72) = 9
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