Greatest Common Factor Calculator

What this calculator does

The Greatest Common Factor Calculator finds the largest positive whole number that divides every number in a list without a remainder. It also shows the prime factorization of each input so you can compare the shared factors.

The greatest common factor is also called the greatest common divisor, or GCD. It is used to simplify fractions, group items evenly, and solve number-theory problems. For two numbers, the GCF is related to the least common multiple by the equation GCF(a, b) = a × b ÷ LCM(a, b).

How to use this calculator

1. Enter two or more whole numbers separated by commas, spaces, or semicolons.
2. Click Calculate.
3. Read the GCF and the prime-factorization steps.

The default example is 330, 75, 450, 225. The greatest common factor is 15.

How the greatest common factor is calculated

The calculator uses the Euclidean algorithm for each pair of numbers. It repeatedly replaces the larger number with the remainder of the division until the remainder is zero. The last non-zero remainder is the GCF.

For the default example, the prime factorizations are:

330 = 2 × 3 × 5 × 11
75  = 3 × 5 × 5
450 = 2 × 3 × 3 × 5 × 5
225 = 3 × 3 × 5 × 5

GCF = 3 × 5 = 15

Only the primes that appear in every factorization are included, and each shared prime is used the smallest number of times it appears in any input.

What the results mean

The result is the largest number that divides every input evenly. In the default example, 330 ÷ 15 = 22, 75 ÷ 15 = 5, 450 ÷ 15 = 30, and 225 ÷ 15 = 15. All four divisions are exact.

Two numbers with a GCF of 1 are called coprime or relatively prime. For example, 8 and 15 are coprime even though neither number is prime.

Assumptions and limitations

• Enter between two and ten positive whole numbers.
• Each number must be 1,000,000,000 or less.
• The calculator returns a positive GCF. It does not return negative common factors.
• Duplicate values are allowed; they do not change the result.
• The calculator shows prime-factorization steps, but it does not draw a factor tree.

Frequently asked questions

What is the difference between GCF and LCM?

The GCF is the largest number that divides every input evenly. The LCM is the smallest number that every input divides evenly. For 12 and 18, the GCF is 6 and the LCM is 36.

Can the GCF be larger than the smallest number?

No. A factor of a number cannot be larger than the number itself, so the GCF cannot exceed the smallest value in the list.

What does it mean if the GCF is 1?

The numbers are relatively prime. They share no positive factor other than 1. This does not mean that the numbers themselves are prime; 8 and 15 are relatively prime, but 8 is composite.

Why is the GCF useful for fractions?

Dividing the numerator and denominator by their GCF reduces a fraction to lowest terms in one step. For example, 18/24 has a GCF of 6, so it simplifies to 3/4.

Source

Reference: Calculator.net — Greatest Common Factor Calculator.