GCF Calculator: Find Greatest Common Factor

A GCF Calculator is a free mathematical tool which helps you calculate the Greatest Common Factor (GCF) for 2 or more numbers. This is the biggest positive integer which can be divided into all the provided numbers without leaving a remainder. It is also known as the Greatest Common Divisor (GCD) or Highest Common Factor (HCF).

For example, the factors of 12 are 1, 2, 3, 4, 6, 12. The factors for 18 are 1, 2, 3, 6, 9, 18. Then we can see that the highest number that goes into both numbers is 6. Therefore 6 is the GCF.

Finding the Greatest Common Factor is important for many areas of mathematics, such as simplifying fractions, solving algebra problems, ratios and understanding relationships between numbers.

However, calculating the GCF manually can be tedious for larger numbers. This online calculator makes things much easier as it gives you the highest common factor.

This can help students, teachers and anyone solving day-to-day maths problems.

How to Use the GCF Calculator

To use this GCF calculator it is as simple as:

Enter the first number

Enter the second number and any additional numbers

Make sure the numbers are entered correctly and there are no any letters or incorrect characters.

Click calculate to calculate the Greatest Common Factor.

The calculator will give you the GCF of all the numbers.

Use the answer to simplify fractions, solve problems or do further calculations.

Example

Let’s say we want to find the GCF of 24 and 36.

First Number: 24

Second Number: 36

The calculator will determine the largest number that can divide into both numbers without leaving a remainder.

The answer will be:
12
GCF = 12

You can also use the calculator to find the GCF of 3 or more numbers.

GCF Calculator Formula

There are many mathematical methods to help calculate the Greatest Common Factor. The most commonly used are prime factorisation and Euclidean algorithm.

Prime Factorisation Method

Prime factorisation involves breaking down each number to find its prime factors.

The formula is:
GCF(a,b)=∏ p i min (m i ,n i )
Where:
\\(a\\) and \\(b\\) are numbers.
\\(p_i\\) are the prime factors in common.
\\(m_i\\) and \\(n_i\\) are exponents.
\\(\\min\\) means choose the smallest exponent.
For example:
24=2^3 x3
36=2^2 x3 2
Common factors are:
2 2 x3 = 12
GCF(24,36)=12
Euclidean Algorithm
The Euclidean algorithm is a quick way to work out the GCF.
The formula is:
GCF(a,b)= GCF(b,a mod b)
Keep doing this until the remainder is 0. The last non-zero remainder will be the GCF.
For example:
36 ÷ 24 = 1 remainder 12
24 ÷ 12 = 2 remainder 0
GCF(24,36) = 12

With 3+ numbers, work out the GCF of the first 2 numbers and then work out the GCF of the resulting answer and the third number. Repeat this until all numbers are included.

Examples

Let’s look at an example of how to calculate the Greatest Common Factor.

Example 1 – GCF of 12 and 20
Given:
First Number = 12
Second Number = 20
Step 1 – Find the factors.
Factors of 12:
1, 2, 3, 4, 6, 12
Factors of 20:
1, 2, 4, 5, 10, 20
Step 2 – Find factors in common.
Common factors: 1, 2, 4
Step 3 – Find the largest number.
GCF(12,20)=4
Answer: 4
Example 2 – GCF of 48 and 60
Prime Factorisation:
48=2^4 x3
60=2^2 x3x5
Common factors:
2 2 x3 = 4x3=12
Answer: 12
Example 3 – GCF of 18, 30 and 42
Find GCF of 18 and 30.
GCF(18,30)=6
Find GCF of 6 and 42.
GCF(6,42)=6
GCF(18,30,42)=6
Answer: 6

Interpreting the Result

Your result will give you the largest positive integer able to divide all the numbers you have inputted.

Understanding your result will help you use it correctly.

GCF – The largest positive integer that can be divided into all of your numbers without leaving a remainder.

Common Factors – Numbers which can be divided into 2 or more values. Example: 2 and 4 are common factors of 12 and 20.

Prime Factors – Prime numbers that can be multiplied together to make the number. For example: 12 can be broken down into 2x2x3.

Simplifying – When a fraction cannot be simplified any further as there are no common factors between the numerator and denominator.

GCF of 1 – When 2 numbers don’t have any common positive factors other than 1 the GCF will be 1. These numbers are referred to as relatively prime or coprime.

For example: GCF of 8 and 15 is 1 as they don’t have any common factors larger than 1.

Factors affecting the calculation

Many mathematical factors can impact the result.

Size of the numbers – The larger the number the more factors it may have which makes it more difficult to work out manually. However, efficient algorithms can calculate the GCF quickly.

Common Prime Factors – The GCF depends on the prime factors common to all your numbers.

Number of inputs – With 3+ numbers the GCF must be divisible by all inputs.

Prime Numbers – If 2 different prime numbers are inputted, the GCF will always be 1.

Even/Odd Numbers – 2 even numbers may have 2 as a common factor. An even and odd number may or may not have common factors.

Zero Values – For a positive integer \\(b\\)0 the GCF of \\(b\\)1 and 0 is \\(b\\)2. However the GCF of 0 and 0 is undefined using the elementary definition.

Negative Numbers – GCF is usually written as positive even if some of the inputs are negative.

Advantages and Real Life Applications

A GCF Calculator can be very useful.

• Quick – Gives you the Greatest Common Factor in seconds.
• Simplify Fractions – Using the GCF will help you simplify fractions. Example: 18/24 can be simplified to 3/4 after dividing the numbers by 6.
• Mathematics – Students can use a GCF Calculator to help them learn about factors, prime numbers, division and how numbers relate.
• Algebra Problems – GCF can help you factorise expressions in algebra.
• Simplifying Ratios – GCF can be used to simplify ratios. Divide the values by the GCF to get the ratio in its simplest form.
• Problem Solving – GCF can be used to divide items into equal groups.
• Accurate – Using automatic calculations ensures accuracy.
• Multiple Numbers – Can calculate GCF for multiple integers.

Accuracy, Assumptions and Limitations

As long as you enter integer values the GCF Calculator will provide an accurate result.

However:

• Integers – GCF is usually calculated using integers.
• Positive – The GCF is usually a non-negative/positive integer.
• Zero – The GCF of a positive integer and 0 will always be the positive integer. The GCF of 0 and 0 is undefined.
• Negative Numbers – Negative numbers can be used by taking the absolute value.
• Prime Numbers – If 2 different primes are inputted, the GCF will always be 
• Multiple Inputs – The GCF must divide all your numbers evenly.
• Large Numbers – If the numbers are very large it may require more computing power depending on the size of numbers the calculator can take.

For the most accurate results enter whole numbers and make sure you have all the required values.

FAQs

What is a GCF Calculator?

A GCF Calculator is an online mathematical tool which calculates the Greatest Common Factor for 2 or more integers. It finds the largest positive number which can be divided into all the numbers without leaving a remainder.

How do I calculate the GCF?

List the factors of each number and find the common factors. Pick the largest one. Prime factorisation and Euclidean algorithm can also be used.

What is the difference between GCF, GCD and HCF?

GCF stands for Greatest Common Factor, GCD is Greatest Common Divisor and HCF is Highest Common Factor. They all mean the same thing.

Can I find the GCF of 3 or more numbers?

Yes! You can find the GCF for 3 or more integers by finding the common factor in all the numbers. For example: GCF of 18, 30 and 42 is 6.

What is the GCF of 2 relatively prime numbers?

If the numbers are relatively prime/coprime the GCF will be 1. For example 8 and 15 don’t have any common positive factors so the GCF is 1.