Math Calculators
GCF Calculator

# GCF Calculator

Greatest common factor calculator finds the GCF of a set of numbers and all factors of these numbers. Solutions with steps for finding the GCF are also demonstrated.

Result

GCF = 4

There was an error with your calculation. ## The Greatest Common Factor Calculator

The greatest common factor calculator is an online tool that allows you to quickly and accurately find the greatest common factor (GCF) of a list of numbers. It will also provide all the coefficients of the numbers in that list.

The GCF is sometimes referred to as the greatest common denominator, the greatest common divisor, or the highest common factor. This GCF calculator can, therefore, be used to find the solution for any of those terms.

## Directions for Use

To use the GCF finder, enter all numbers separated by commas or spaces and press “Calculate.” The calculator will return the GCF of the listed numbers and will demonstrate the solution for finding its value. The calculator will always illustrate the solution by factorization.

Limitations on the input values

1. You must input whole numbers.
2. Only one of the numbers can be zero.
3. You can only enter positive integers.

## The Definition of the Greatest Common Factor

In mathematics, the greatest common factor of a list of positive integers is the largest positive integer that divides all the numbers in the list without remainders.

So, what is a factor? A number’s factors, or divisors, are integers that can divide the given number without any remainder.

For example, the factors of number 12 would be 1, 2, 3, 4, 6, and 12. The common factors of several numbers are those factors that can divide all those numbers without remainders. For example, if we had to find all common factors of numbers 12 and 16, we would first need to list all the factors of each number and then check which factors are on both lists:

12: 1, 2, 3, 4, 6, 12

16: 1, 2, 4, 8, 16

The common factors of the given numbers (12 and 16) are 1, 2, and 4. The greatest common factor is simply the largest one of these numbers. In the case of 12 and 16, the GCF is 4.

## How to Find the Greatest Common Factor

There are several ways to find the GCF of several numbers. The most straightforward one is the solution by factorization.

### Solution by factorization

To find the GCF using this method, follow the steps described above — first, identify the factors of all the numbers in the list, then find the common factors and choose the largest one.

### Calculation example

Find the greatest common factor of the numbers 3, 9, and 48.

Solution:

• The factors of 3 are 1, and 3.
• The factors of 9 are 1, 3, and 9.
• The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

Common factors are 1 and 3. Then the greatest common factor is 3.

### Prime factorization

Another strategy for finding the greatest common factor of a set of numbers consists of the following steps:

1. Find all the prime factors of the numbers in the given set.
2. List the common prime factors for all set numbers.
3. To get the greatest common factor, multiply the common prime factors.

## Calculation example

Find the greatest common factor of the numbers 16, 24, and 76.

Solution

• The prime factorization of 16 is: 2 × 2 × 2 × 2, or 2⁴.
• The prime factorization of 24 is: 2 × 2 × 2 × 3, or 2³ × 3¹.
• The prime factorization of 76 is: 2 × 2 × 19, or 2² × 19¹.
• The common prime factors are: 2 × 2, or 2².

Therefore, the greatest common factor is: 2 × 2 = 2² = 4

## Euclid’s algorithm

This algorithm is handy for finding the greatest common factors of large numbers, where using any type of factorization would be very cumbersome and time-consuming. This algorithm, developed by Euclid, uses the fact that the GCF of the two numbers m and n, where m > n, is the same as the GCF of the two numbers n and m - n.

To use this algorithm for the finding the GCF of the two numbers m and n, you need to consecutively replace the largest of the two numbers with the difference of the numbers:

First, replace m with m - n. Now you have a new set of numbers: m - n and n.

Check which of the numbers is larger, and replace that number with the difference between the current numbers.

Repeat until the two numbers become equal. That number will be the greatest common factor of the original set of numbers.

## Calculation example

Find the greatest common factor of the following numbers: 124, 98.

Solution

The larger number in this set is 124. Let’s replace it with the difference in numbers 124 - 98 = 26 so that we get the following set:

26, 98

The larger number in this set is 98. Let’s replace it with the difference in numbers, (98 - 26) = 72 so that we get the following set:

26, 72

We can subtract 26 from the larger number two more times: 72 - 26 - 26 = 20. Now our set looks like this:

26, 20

In the following iteration, we replace 26 with 26 - 20 = 6 to get

6, 20

Next, we subtract 6 from 20. We can repeat this operation three times since the resulting difference will still be greater than 6:

20 - 6 - 6 - 6 = 2

Now our set is:

6, 2

The following iterations are:

(6 - 2 = 4), 2 or 4, 2

(4 - 2 = 2), 2 or 2, 2

Now we have a set of two equal numbers:

2, 2

Therefore, the greatest common factor of 124 and 98 is 2.