How to Find the GCF (Greatest Common Factor)

Whether you are trying to simplify fractions in algebra or divide a space evenly into a grid, you need to know how to find the Greatest Common Factor (GCF).

Also known as the Greatest Common Divisor (GCD), the GCF is simply the largest number that can divide evenly into a set of numbers without leaving a remainder.

Below, we have built an interactive engine that will not only find the GCF of any two numbers for you, but will actually show you the step-by-step mathematical work using the famous Euclidean Algorithm!

The Interactive GCF Calculator

Enter any two whole numbers below. The engine will instantly calculate the GCF and generate the mathematical steps required to find it.

Method 1: The List Method (Best for Small Numbers)

If you are dealing with relatively small numbers (like 12 and 18), the easiest way to find the GCF is simply to list out all of their factors and find the biggest match.

Example: Find the GCF of 12 and 18.

1. List the factors of 12: The numbers that divide evenly into 12 are 1, 2, 3, 4, 6, and 12.
2. List the factors of 18: The numbers that divide evenly into 18 are 1, 2, 3, 6, 9, and 18.
3. Find the largest match: The numbers they have in common are 1, 2, 3, and 6. The largest one is 6.

Therefore, the GCF of 12 and 18 is 6.

Method 2: The Euclidean Algorithm (Best for Large Numbers)

If you try to use the List Method for massive numbers like 1,071 and 462, you will be writing factors for hours.

Over 2,000 years ago, the Greek mathematician Euclid invented a brilliant, highly efficient algorithm for finding the GCF of massive numbers. It relies on a simple principle: The GCF of two numbers also divides their difference.

The algorithm works through a series of remainders.

Example: Find the GCF of 1,071 and 462.

1. Divide the larger by the smaller: Divide 1,071 by 462.
   • 462 goes into 1,071 two times (462 × 2 = 924).
   • The remainder is 147 (1,071 - 924).
2. Shift the numbers: Now, take the old divisor (462) and divide it by the new remainder (147).
   • 147 goes into 462 three times (147 × 3 = 441).
   • The remainder is 21 (462 - 441).
3. Shift the numbers again: Take the old divisor (147) and divide it by the new remainder (21).
   • 21 goes into 147 exactly seven times (21 × 7 = 147).
   • The remainder is exactly 0.

Because the remainder is now 0, we stop. The last non-zero remainder we had was 21.

Therefore, the GCF of 1,071 and 462 is 21.

Our interactive calculator above actually runs this exact Euclidean Algorithm in the browser to give you your answers in milliseconds!