Finding the Greatest Common Divisor or Greatest Common Factor
You probably learned about Greatest Common Divisor (GCD) or Greatest Common Factor (GCF) in junior high, and it was probably in the context of simplifying fractions. For example, to simplify 
, one could see that 40 and 60 are each divisible by 2, resulting in the fraction 
, but this fraction needs to be simplified further. Again, 20 and 30 are each divisible by 2, resulting in the fraction 
. We still haven’t simplified this fraction and we’ve already reduced it twice! If we knew the GCF of 40 and 60, we could simplify this fraction in one step. Note that 40 and 60 are each divisible by 20, so that 
. How do we find the greatest common factor of 40 and 60? We’ll find out in Example 1 below.
What is a GCF or GCD?
Let x and y be two nonzero whole numbers. The greatest common factor of x and y is defined to be the largest whole number that is a factor of both x and y. This is written as gcf (x, y) or GCF (x, y). Some books use greatest common divisor instead of factor. This is written as gcd (x, y) or GCD (x, y).
For example:
a) gcd (6,15) = 3
b) GCF (16, 12) = 4
c) gcf (10, 100) = 10
If you have more than two nonzero whole numbers, then the greatest common factor of all the numbers is the largest whole number that is a factor of each number.
Example 1
Find the GCD(60, 40):
Solution 1
Step 1: Find the prime factorization of 60 and 40. Use factor trees or any method you prefer.
Step 2: Check to see which factors they have in common. Note that both numbers have two factors of 2 and one factor of 5. The product of these common prime factors will give us the GCD.
Example 2
Find the gcf (18, 30, 54).
Solution 2
Step 1: Find the prime factorization of 18, 30 and 54
30 = 
54 = 
Step 2: Each number has one factor of 2 and one factor of 3.
gcf (18, 30, 54) = 2 · 3 = 6
Notice that 18 = 6 • 3
30 = 6 • 5
54 = 6 • 9
and that the numbers 3, 5, and 9 have no common factors larger than 1. This proves that we have found the gcf of 18, 30, and 54 which is 6.