# Greatest Common Factor Calculator

Find the greatest common factor, common factors, and all factors for a set of numbers using the calculator below.

## Solution:

**22**

**22**

## How to Find the Greatest Common Factor of a Set of Numbers

The *greatest common factor (GCF)* of a set of numbers is the largest factor that is common to all of the numbers in the set. The greatest common factor is sometimes referred to as the *highest common factor (HCF)*, *greatest common denominator (GCD)*, *greatest common divisor*, or *highest common divisor*.

A *factor* of a number x is a number that can be multiplied by another whole number to get x.

There are a few methods for finding the greatest common factor. The first is to use the calculator above, which even shows all of the steps. Follow along below to learn about three more methods.

### Method One: Find GCF Using Prime Factorization

One method to finding the greatest common factor of a set of numbers is to use prime factorization. To use prime factorization, find the prime factors of each number.

To find the prime factors, divide each number in the set by a prime factor. Then divide each number by another prime factor until they cannot be divided further. The prime numbers used to divide the number are the prime factors.

To find the greatest common factor of 90 and 165 using prime factorization, find the prime factors of each number.

To find the prime factors, divide each number by another prime factor until they cannot be divided further.

When you have the prime factors for each number in the set, the next step is to find the numbers that are common to all of the numbers.

Finally, multiply all of the common prime factors together to find the greatest common factor.

**For example,** let’s find the greatest common factor of 90 and 165 using prime factorization.

To find the prime factors of 90, divide 90 by 2 to get 45. 2 is a prime number but 45 can be divided by 5 to get 9. 5 is a prime number but 9 can be divided by 3 to get 3.

Thus, the prime numbers of 90 are **[5,3,3,2]**.

Repeat this process for 165. 165 divided by 3 = 55. 3 is a prime factor but 55 can be divided by 5 to get 11.

The prime factors of 165 are **[11,5,3]**.

Now, find the prime factors that are common to both numbers. **5** and **3** are in both sets of prime factors we found above.

To find the greatest common factor, let’s multiply the common prime factors together.

5 × 3 = 15

Thus, **15** is the greatest common factor of 90 and 165.

Here’s a tip: use our prime factors calculator to find all prime factors of a number.

### Method Two: Find GCF By Finding All Factors

Using this method, the greatest common factor is found by finding all of the factors for each number in the set, then finding the factors common to all of the numbers in the set. The largest factor common to all numbers is the greatest common factor.

To find the factors of a number, divide by 2, if the result is a whole number than both 2 and the result are factors. Repeat this process by dividing by 3, 4, and so on until you’ve reached 1.

Then, find the factors that are in the set of factors for all numbers. These are the common factors of the numbers.

Finally, find the largest number in the set of common factors. This is the greatest common factor.

**For example,** let’s find the greatest common factor of 90 and 165 by finding all factors.

Let’s start by finding the factors for both numbers.

The factors for 90 are **[1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90]**

The factors for 165 are **[1, 3, 5, 11, 15, 33, 55, 165]**

Next, find the factors that are common to both 90 and 165.

The common factors are **[1, 3, 5, 15]**

Find the largest number in the set of common factors to find the greatest common factor.

The largest number in the set of common factors of 90 and 165 is 15. Thus, the greatest common factor of 90 and 165 is **15**.

### Method Three: Find GCF Using Euclid’s Algorithm

Euclid’s algorithm defines the steps to efficiently find the greatest common factor of two numbers. Start by dividing the large number by the small number to get the quotient and remainder.

If the remainder is not 0, then divide the previous divisor by the remainder. Continue doing this as long as the remainder is not 0. Once you have a remainder that equals 0 the divisor in that step is the greatest common factor.

**For example,** let’s find the greatest common factor of 90 and 165 using Euclid’s algorithm.

165 ÷ 90 = 1 remainder 75

90 ÷ 75 = 1 remainder 15

75 ÷ 15 = 5 remainder 0

Because 15 is the divisor in the final step when the remainder equals 0, the GCF is **15**.

Learn more using our Euclidean algorithm calculator.

## What is the Greatest Common Factor Used For

The greatest common factor is most commonly used to simplify or reduce fractions. The greatest common factor is also known as the greatest common divisor. Learn more about reducing fractions using our reducing fractions calculator.