Greatest Common Factors
In Math . . .a is a common factor of b and c if a is a factor of both b and c. a is the greatest common factor of the two numbers if there are no common factors greater than a. We'll write "the greatest common factor of a and b" as gcf(a, b).  In English . . .Both phrases mean exactly what they seem to say: A number is a common factor of two other numbers if it's a factor of both of them. So, for example, 2 is a common factor of 12 and 18 but 4 isn't a common factor since it's a factor of 12 only. A little experimentation will show you that 3 and 6 are the only other factors that 12 and 18 have in common. Because 6 is the biggest of the common factors, it would be the greatest common factor of 12 and 18. 
Quick Tip  Which One Did You Find?
At this point, I think it's worth repeating that factors are always smaller than the original number and multiples are always larger.
In our previous example, finding the greatest common factor was relatively straightforward since the numbers were small. It was easy enough to write out all the factors and pick the largest common one from the two lists. For larger numbers that approach would be impractical. Here's a procedure that will let you find the greatest common factor even for large numbers.
 Write out the prime factorizations of both numbers.
 List all the factors that they have in common.
 If a factor appears more than once, add it to your list the smaller number of times.
 Multiply the numbers in your list together and the result will be the greatest common factor.
If you're comfortable working with exponents they can make the last step of the procedure a little clearer.
 Write out the prime factorizations of both numbers using exponents.
 List all the factors that they have in common.
 Give each factor the smallest exponent from the two factorizations.
 Multiply the resulting numbers together.
Examples
Example 1Find the greatest common factor of 12 and 18.
Example 3Find the greatest common factor of 100 and 80. We'll work this one out using the exponent method.

Example 2Find the greatest common factor of 18 and 36.
Example 4Find the gcf(495, 945) Here's another method for finding the greatest common factor that's a little more visual. I'm going to start by writing out the prime factorizations with the common factors lined up.
To find the greatest common factor, I just took the numbers from each column that appeared in both of the rows and multiplied them together. 
Videos

Dynamic Tutorial  Finding the Greatest Common Factor
Directions: This solution has four steps. To see a description of each step click on the boxes on the left side below. To see the calculations, click on the corresponding box on the right side. Try working out the solution yourself and use the descriptions if you need a hint and the calculations to check your solution.