What is a common factor?

A common factor (also called a common divisor) is a whole number that divides evenly into two or more other numbers with no remainder. For example, 4 is a common factor of 12 and 20 because 12 divided by 4 equals 3 and 20 divided by 4 equals 5, both with no remainder. The greatest common factor (GCF) is the largest number that divides evenly into all of the given numbers. The GCF of 12 and 20 is 4.

How do you find the greatest common factor (GCF)?

The most efficient method for finding the GCF is the Euclidean algorithm. To find the GCF of two numbers, divide the larger number by the smaller and take the remainder. Then divide the smaller number by that remainder, and continue until the remainder is zero. The last non-zero remainder is the GCF. For example, GCF(48, 18): 48 divided by 18 gives remainder 12; 18 divided by 12 gives remainder 6; 12 divided by 6 gives remainder 0; so GCF = 6. Once you have the GCF, every divisor of the GCF is also a common factor of the original numbers.

What is the difference between GCF and LCM?

The GCF (greatest common factor) is the largest number that divides evenly into all of the given numbers. The LCM (least common multiple) is the smallest number that all of the given numbers divide evenly into. They are related by the formula: GCF(a, b) x LCM(a, b) = a x b. For example, for 12 and 18: GCF = 6 and LCM = 36; and 6 x 36 = 216 = 12 x 18. GCF is used when simplifying fractions; LCM is used when adding or subtracting fractions.

Common Factor Calculator

Enter two or more whole numbers to find all their common factors, the greatest common factor (GCF), and the least common multiple (LCM). Step-by-step working is shown for each calculation.

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Standard method Uses the Euclidean algorithm for GCF. Results are exact for all positive integers.

1. Enter Your Numbers

First number

Second number

Third number (optional)

Fourth number (optional)

2. Results

Factors of 48

Factors of 36

Factors of

Common Factor Results

Greatest Common Factor
12
GCF / HCF

Number of Common Factors

Shared divisors

Least Common Multiple

144

LCM

All Common Factors

1, 2, 3, 4, 6, 12

Every shared divisor

Step-by-Step Working

Factor Summary

Numbers entered48, 36

Factors of first number-

Factors of second number-

All common factors-

Greatest Common Factor (GCF)-

Fraction Simplification

Original fraction (a/b)48/36

GCF to divide by-

Simplified fraction-

Least Common Multiple-

GCF x LCM = a x b-

What Are Common Factors?

A factor of a number is any whole number that divides it evenly with no remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, because each of these divides into 12 without leaving a remainder. A common factor of two or more numbers is a factor that appears in the factor list of every one of those numbers. The greatest common factor (GCF) is simply the largest of these shared factors.

The GCF is also called the highest common factor (HCF) or greatest common divisor (GCD) depending on the country and context. In New Zealand, both GCF and HCF are used in mathematics education.

How This Calculator Works

This calculator uses the Euclidean algorithm to find the GCF of the numbers you enter. Once the GCF is known, every divisor of the GCF is also a common factor of the original numbers. The steps are:

Find the GCF of the first two numbers using the Euclidean algorithm.
If more numbers are entered, find the GCF of that result and the next number, repeating until all numbers are included.
List all factors of the final GCF. These are all the common factors of the original numbers.
Calculate the LCM using the relationship: LCM(a, b) = (a x b) / GCF(a, b).

The Euclidean Algorithm

The Euclidean algorithm is one of the oldest known algorithms, described by the Greek mathematician Euclid around 300 BC. It finds the GCF of two numbers by repeated division:

Divide the larger number by the smaller number and find the remainder.
Replace the larger number with the smaller number, and the smaller number with the remainder.
Repeat until the remainder is zero. The last non-zero remainder is the GCF.

Example: GCF(48, 36). Step 1: 48 divided by 36 gives remainder 12. Step 2: 36 divided by 12 gives remainder 0. The GCF is 12.

Worked Example: Matching the Default Inputs

For the numbers 48 and 36:

Step	Division	Remainder
1	48 / 36	12
2	36 / 12	0

GCF = 12. The factors of 12 are: 1, 2, 3, 4, 6, 12. All six of these are common factors of 48 and 36. The LCM = (48 x 36) / 12 = 1,728 / 12 = 144. The fraction 48/36 simplifies to 4/3 (dividing both by the GCF of 12).

Why Common Factors Matter

Common factors are used throughout mathematics:

Simplifying fractions: Divide both numerator and denominator by their GCF to get the simplest form. For example, 48/36 becomes 4/3.
Adding and subtracting fractions: Find the LCM of the denominators to find the lowest common denominator.
Factoring algebraic expressions: The GCF is factored out when simplifying polynomials.
Solving problems involving groups: If you need to split objects into equal groups, the GCF tells you the largest possible group size.

Related Calculators

Maths and Stats Calculators: full collection of maths tools.
Combination (nCr) Calculator: calculate combinations from a set.
Prime Number Checker: test whether a number is prime.
Adding Fractions Calculator: add fractions using the LCM of denominators.
Quadratic Formula Solver: solve quadratic equations with step-by-step working.

Method: GCF calculated using the Euclidean algorithm (division form). LCM derived from the identity LCM(a, b) = (a x b) / GCF(a, b), extended to multiple numbers by iteration. All factors of the GCF are enumerated by trial division from 1 to sqrt(GCF).

This calculator works with positive whole numbers only. Inputs are rounded to the nearest integer. For very large numbers (above 10,000,000), results remain exact as the Euclidean algorithm is not affected by number size.