Greatest Common Factor

The Greatest Common Factor (GCF) is the largest positive integer that divides two or more non-zero integers without a remainder.

In simpler terms, the greatest common factor of two or more whole numbers is the largest whole number that divides evenly into each of the numbers.

Still Confused? Let’s break it down first.

GCF Calculator - Factor Bar

A Factor is any number that can be multiplied to make another number. For example:
GCF Calculator - Factor Example

In this equation, 6 is the number you eventually get (called the “product”), but in order to get that number, we have to take the factors, 2 and 3, and multiply them. The number 2 can divide 6 evenly; there is no remainder. The number 3 can also divide 6 without a remainder. So we can say that the factors of the number 6, are numbers that divide into 6 evenly, and produce no remainder.

The factors 2 and 3 in this case are also called prime numbers. A prime number is a number that has exactly two factors, 1 and itself. Why is this information necessary, and why is it relevant? The greatest common factor can be solved many different ways (not to mention, the GCF Calculator on this page can solve the GCF as well), but here at GCFcalculator.com, we will show you how to solve the greatest common factor using prime factorization.

GCF Calculator - Prime Factorization Bar

Prime factorization can be done using graphical representation in tree form (factor trees), showing the prime factors of a composite number. Using a factor tree, we can find the prime factorization of 36.

GCF Calculator - Factor Tree of 36

The factor tree is accomplished by finding factors for our composite number, then expanding down each other non-prime number in the tree until we have nothing left but prime numbers. In this example, 18 and 6 are not prime numbers, so we continue to expand them as well. Although it is not necessary, we circled the prime numbers we have found to help us visually see what we have done. We can check if our tree is correct by multiplying the prime numbers together, and see if it equals our composite number.
GCF Calculator - Prime Factorization
GCF Calculator - Greatest Common Factor Bar


Solving for the Greatest Common Factor by hand can be done without using the GCF Calculator, and without much trouble. Let’s find the GCF of 36 and 28, by first finding the prime factors of each number, using prime factorization like before.

As you see above, the prime factorization of 36 is 2 x 2 x 3 x 3. Now we use the same method to find the prime factors of 28.
GCF Calculator - Factor Tree of 28

Here the prime factorization of 28 is 2 x 2 x 7. Looking at both of these trees, we now compare and identify those prime factors that both numbers have in common, and we multiply them. We see that the number 2 is circled twice in each tree. This means our common factors are 2 and 2.
GCF Calculator - GCF

Multiplying 2 times 2, gives us 4. Therefore, the Greatest Common factor of 36 and 24 is 4. Use the GCF Calculator to check and see the answer for yourself! Don’t forget to watch the video below for more help without using the GCF Calculator. Remember, the more you practice solving the Greatest Common Factor by hand, the less you will rely on tools such as the GCF Calculator to do the work for you.