Greatest Common Divisor Calculator (GCD)

Enter a list of integers to calculate their greatest common factor. Enter the numbers separated by a comma.

Whole numbers


Step-by-step solution:

With this calculator, you can calculate the greatest common divisor (GCD) of a list of numbers. The numbers must be entered separated by a comma. You can calculate the GCD of any list of numbers.

How to use the greatest common divisor calculator?

Step 1: Enter the numbers in the corresponding box. You can enter a list of multiple numbers. However, the numbers must be integers and must be separated by a comma. For example, to calculate the GCD of 3, 6, and 8, simply enter 3,6,8.

Step 2: Click “Calculate” to get the greatest common divisor of the numbers.

Step 3: The GCD will be displayed on the right side. In addition, the step-by-step solution through the prime factorization method will be displayed at the bottom of the calculator.

What is the greatest common divisor?

The greatest common divisor of two numbers is the largest integer that divides both numbers without leaving a remainder. It is the largest multiple of both numbers. For example, the greatest common divisor of 20 and 15 is 5, since 5 divides both numbers without leaving a remainder and there is no larger number that does this.

How to find the greatest common divisor?

There are several methods we can use to find the GCD of two or more numbers. Two of the most important methods are by factor lists and by prime factorization.

Finding the GCD using factor lists

We can write a list of all the factors for the given numbers. For example, suppose we have 12, 24, and 30. Their factors are:

  • 12: 2, 3, 4, 6, 12
  • 24: 2, 3, 4, 6, 8, 12, 24
  • 30: 2, 3, 5, 6, 10, 15, 30

Now, we determine the largest number that is common to all three numbers. In this case, it is 6. Therefore, the GCD of 12, 24, and 30 is 6.

Finding the GCD using prime factorization

We can write the prime factorization of each of the given numbers. Prime factorization consists of writing the prime factors of the number so that when we multiply these factors, we get the original number. For example, taking 12, 24, and 30 again, we have:

  • 12: 2, 2, 3
  • 24: 2, 2, 2, 3
  • 30: 2, 3, 5

Now, we take every prime factor that is common to all numbers. If a factor is repeated more than once in each number, we write it the number of times it is repeated.

In this case, only 2 and 3 are common. By multiplying these numbers, we get 6, which is the GCD of the given numbers.

Why find the greatest common divisor?

Calculating the greatest common divisor has some applications. One of the most frequent applications is to simplify fractions. For example, if we have 20/15, we know that 5 is the greatest common divisor of 20 and 15, so we divide by 5, and we get 4/3, which is a fraction that can no longer be simplified.

Is it possible to calculate the greatest common divisor of more than two numbers?

The greatest common divisor can be calculated for any list of numbers. You can use the calculator above to get the GCD of a list of multiple numbers. The process for calculating the GCD is the same as for two numbers.

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