Common Factor Calculator

Background.

Two numbers share a common factor when both can be divided by it with nothing left over, and the largest of those shared divisors is the one most practical questions need. It is the number that cuts 84/126 down to 2/3 in a single step. It is the largest number of identical packs you can make from 84 red tiles and 126 blue ones with no tile left over. Divide it into the product of the two numbers and you get their least common multiple, the figure behind common denominators and repeating schedules.

School usually teaches two ways to find it: list every factor of both numbers and pick the largest one they share, or break each number into primes and multiply the primes they have in common. Both work for small numbers and both become painful quickly; factoring a fifteen-digit number by hand is not realistic.

This page uses the method Euclid set down more than two thousand years ago, which needs no factoring at all. Divide the larger number by the smaller and keep only the remainder. Then divide the previous divisor by that remainder, and repeat. The remainders shrink every time, and the last divisor before a remainder of zero is the greatest common factor. It works because anything that divides both numbers must also divide their difference, and so their remainder, which lets the pair be swapped for a smaller pair with exactly the same common factors.

The method stays quick even for huge inputs, since the number of division steps never exceeds five times the number of digits in the smaller number. Enter two positive whole numbers up to 9,000,000,000,000,000; decimals, zero and negatives are refused, and three or more numbers are handled two at a time.

What is common factor calculator?

The greatest common factor (GCF) of two positive integers, also called the greatest common divisor or highest common factor, is the largest whole number that divides both of them exactly. Every other common factor of the pair is itself a factor of the GCF: the common factors of 84 and 126 are 1, 2, 3, 6, 7, 14, 21 and 42, all of them divisors of 42. When the GCF is 1 the two numbers are called coprime, or relatively prime, even if neither is a prime number.

How to use this calculator.

1. 01Enter the First positive integer and the Second positive integer in either order; swapping them gives the same answer.
2. 02Use whole numbers only. For decimals such as 1.2 and 1.8, multiply both by the same power of ten to get 12 and 18, find their GCF of 6, then divide back to 0.6.
3. 03For three or more numbers, find the GCF of the first two, enter that result with the third number, and carry on down the list.
4. 04Read the greatest common factor. Divide a fraction's top and bottom by it to put the fraction in lowest terms, or divide the product of the two numbers by it to get their least common multiple.

The formula.

gcd(a, b) = gcd(b, a mod b), repeated until the second number is zero, at which point the first number is the answer. Here a mod b is the remainder left when a is divided by b. The rule holds because any number dividing both a and b also divides a − b, a − 2b and so on, and therefore divides the remainder; in the other direction, anything dividing b and the remainder also divides a. The pair (b, a mod b) therefore has exactly the same common factors as (a, b) but is smaller, and since the remainders keep shrinking, the process must reach zero. If a starts out smaller than b, the first step simply swaps them, because a mod b is then a itself. The arithmetic is exact integer division with nothing to round. Input size barely affects the work: the number of steps grows with the number of digits, not with the size of the numbers.

A worked example.

Frequently asked questions.

References& sources.

1. [1]OpenStax, Rice University. College Algebra 2e, 2021. Algebra and function chapters. Retrieved 2026-08-06.
2. [2]OpenStax, Rice University. Precalculus 2e, 2021. Trigonometry and analytic geometry chapters. Retrieved 2026-08-06.
3. [3]NIST/SEMATECH. e-Handbook of Statistical Methods, 2012. Chapter 1.3 and Chapter 4. Retrieved 2026-08-06.