To understand Greatest Common, we must understand the words:

Once we understand these words, we are ready to understand Greatest Common.  So, lets look at the words and learn their meaning:

Greatest - Biggest, that which is larger than the others.   (example: Given 6, 8, 9 and 11,    11 is the greatest.)

Common - Shared by all, belonging equally to two or more places. (examples: 3, and 9 are common to the lists {1,3,5,7,9,11} and {3,6,9,12,15} ).

Now, let's look at the phase Common Factors.

Greatest Common

Given the sets { 3, 5, 6, 7} and {3, 5, 6, 7}

      7 is the greatest common thing.

 

Given the sets {5,7,9,11,13} and { 5,8,10,11},

       11 is the greatest common thing.

 

Given the sets {25,11,6,21} and {11,21},

       21 is the greatest common thing.

 

Given the sets {1,2,3,6,9,18} and {1,3,9},

       9 is the greatest common thing.