To find the GCF for three or more numbers, follow these steps:
1) Determine which of the given numbers is smallest, then find the smallest difference between any pair of numbers.
2) See what is smaller: the smallest number, or the smallest difference. Whichever one is smallest, that number is the GPGCF (Greatest Possible GCF). That means that this is the biggest number that the GCF could possibly be. Or, more formally we would say: The GCF, if it exists, must be less than or equal to the GPGCF.
3) Check if the GPGCF itself goes into all of the given numbers. If so, then it is the GCF. If not, list the factors of the GPGCF from largest to the smallest and test them until you find the largest one that does divide evenly into the given numbers. The first factor (i.e., the largest factor) that divides evenly into the given numbers is, by definition, the GCF.
Problem: Find the GCF for 18, 30, 54.
1) Note that the smallest number is 18, and the smallest difference between the pairs is 12 [54 – 30 = 24; 54 – 18 = 36; 30 – 18 = 12] .
2) Of those four quantities (the smallest number and the three differences), 12 is the least. This means that the
GPGCF = 12.
3) Check if 12 divides evenly into the three given numbers: 18, 30 and 54. In fact, 12 doesn’t divide evenly into ANY of these numbers. Next we check the factors of 12, in order from largest to smallest. Those factors are: 6, 4, 3 and 2. The first of those that divides evenly into all three numbers is 6. [18 ÷ 6 = 3; 30 ÷ 6 = 5; 54 ÷ 6 = 9]. So the GCF = 6. And we are done.
MORE CHALLENGING PROBLEM:
Find the GCF for 24, 148, 200.
1) Note that the smallest number is 24, and that the smallest difference between the pairs is 52 [200 – 148 = 52; 200 – 24 = 176; 148 – 24 = 124] .
2) Of those four quantities (the smallest number and the three differences), 24 is the least. This means that for this problem, the GPGCF = 24.
3) Check if 24 divides evenly into the three given numbers: 24, 148 and 200. While 24 does divide evenly into 24, it does not divide evenly into 148 or 200. So next we check the factors of 24, in order from largest to smallest. Those factors are: 12, 8, 6, 4, 3 and 2. The first of those that divides evenly into the three given numbers is 4. [24 ÷ 4 = 6; 148 ÷ 4 = 37; 200 ÷ 4 = 50]. So the GCF = 4. And, once again, we are done.
The process may seem a bit long, but once you get used to it and start doing it in your mind, not on paper, you should find that it actually is quite fast. And you’ll find yourself figuring out the GCF for three or more numbers all in your mind — with no need for pencil and paper — while everyone around you will be making prime factor trees or using calculators. And surely that is a good feeling.
Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com Just click the links in the sidebar for more information!