What is the greatest common factor?

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1. What is the greatest common factor? How do you know when you have found the greatest one? Provide an example and a detailed explanation.
The greatest common factor (or GCF) of two (or more) quantities is the largest factor that evenly divides each (or all) of the quantities, with a remainder of 0 in all cases.
Example: the numbers 300 and 360 have many common factors (like 2, 3, 5, 6 and 10 to name a few of their common factors), but the gcf of 300 and 360 is 60.
Strategies for Finding the GCF:
One way to find the gcf may be called the "brute force" method.
Strategy #1 For Finding the Greatest Common Factor (GCF) of Two or More Number (Quantities) Based on the "Brute Force" Method:
Step 1: List all of the factors of each number for which you're trying to find the gcf in separate lists. Step 2: Create another list which contains ony those factors which are common to all of your lists from step 1.
Step 3: From your list in step 2 of all of the common factors, identify the largest of all the common factors. This is the gcf.
The "brute force" method would be applied to 300 and 360 as follows:
Step 1:
Factors of 300 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150 and 300.
Factors of 360 are: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360.
Step 2:
The factors that are common to both of the previous lists are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60.
Step 3:
The largest factor that is listed in step 2 is 60. 60 is the gcf of 300 and 360.
Problems with the brute force method include completing step 1 accurately when the numbers involved are large (it's easy to forget, or miss one or ...