Greatest Common Factor and Least Common Multiple

GCF and LCM

Greatest Common Factor (GCF)

Definition of Greatest Common Factor (GCF) :  

The greatest common factor (GCF) of two natural numbers n and m is the greatest natural number k that is a factor of both n and m. We symbolize this as GCF(n, m) = k.      

Example:  

Relating this definition to the problem we solved on the previous page, we see that the two numbers n and m are the numbers 20 and 16.   We may write the solution symbolically as GCF(20, 16) = 4.   This says "the greatest common factor of 20 and 16 is 4."

The number k in the definition corresponds to the 4 in the problem. It is the factor of the greatest value shared by both n and m. The number k is the Greatest Common Factor of 20 and 16.  

 One method for finding the greatest common factor is to choose the value that is the greatest value in the intersection of the sets of common factors.

Example: Find the GCF(20, 16).

The set of factors of 20 is {1, 2, 4, 5, 10, 20} and the set of factors of 16 is {1, 2, 4, 8, 16}.

{1, 2, 4, 5, 10, 20} ∩ {1, 2, 4, 8, 16} = {1, 2, 4} which has greatest value 4.

So, GCF(20, 16) = 4.

Notice that an intermediate step in this method applied to the problem at the beginning of the session gives all possibilities for packaging all the balls and paddles: one, two, or four packages.

 Example: Find GCF(18, 24).

The set of factors of 18 is {1, 2, 3, 6, 9, 18}
and set of factors of 24 is {1, 2, 3, 4, 6, 8, 12, 24}.

{1, 2, 3, 6, 9, 18} ∩ {1, 2, 3, 4, 6, 8, 12, 24} = {1, 2, 3, 6} which has greatest value 6.

The GCF (18, 24) = 6.

 

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