The Greatest Common Factor (GCF) Calculator takes an input of two to six (in advanced mode) numbers and outputs the greatest common factor among them. That means the largest number that will evenly divide all the input numbers. In the case where no whole integer is possible beyond 1, this means the numbers are relatively prime, or coprimes.

This calculator is most useful for dividing sets of fractions. It also might be useful for other real-world applications where you need to find a common value between different ratios, and other niche problems.

Example

Given the numbers 288, 396, and 144, the GCF is 36. Even though all three numbers have multiple factors in common, including 2 and 3, the highest even divisor common to all of them is 36. Here's how that works:

•	The total divisors of 288 are 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144.
•	The total divisors of 396 are 2, 3, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198.
•	The total divisors of 144 are 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72.
•	The highest number common to each of those lists is 36.

Prime Factors

The total divisors are not to be confused with its prime factorization. We don't care here about every factor being prime, just in finding the highest common divisor between the list of numbers we have entered. However, reducing each number to its prime factors is helpful in also calculating the greatest common factor.

Returning to our previous example…

•	The prime factors of 288 are 2, 2, 2, 2, 2, 3, and 3.
•	The prime factors of 396 are 2 2 3 3 11.
•	The prime factors of 144 are 2, 2, 2, 2, 3, and 3.

You see a pattern in each factorization, where the set { 2, 2, 3, 3 } are common to all three numbers. Multiply 2 * 2 * 3 * 3 and there's your 36!

Euclid's Method

Of course, the clever ancient Greek mathematician Euclid had a shortcut. He noticed that the GCF of two numbers also divides their difference, and used a division algorithm as follows:

•	Find the GCF for 102 and 221.
•	221 / 102 = 2, with a remainder of 17.
•	102 / 17 = 0 with no remainder, so we stop here. 17 is the GCF of 102 and 221.
•	To check the difference, 221 - 102 = 119, and the prime factors of 119 are { 7, 17 }.

Finding the GCF of number sets, or proving them to be coprime, is also a mathematical field that has relevance in cryptography.