Greatest Common Factor (GCF) Calculator

This professional-grade calculator finds the Greatest Common Factor (also called Greatest Common Divisor, GCD) of two or more integers in an instant. It’s built for students, educators, and professionals who need reliable results, complete with Euclidean algorithm steps and accessible, mobile-first UX.

Data Source and Methodology

Authoritative Reference: NIST Digital Library of Mathematical Functions (DLMF), Chapter 1: Elementary Functions, and classical number theory texts on the Euclidean Algorithm.
Reference link: https://dlmf.nist.gov/ (accessed 2025).

Tutti i calcoli si basano rigorosamente sulle formule e sui dati forniti da questa fonte.

Implementation details: We compute the GCF by iteratively applying the Euclidean algorithm across the list: gcd(a, b, c, …) = gcd(gcd(a, b), c, …). For transparency, the tool can show each division step and optionally prime factorizations for small and medium integers.

The Formula Explained

\( \gcd(a,b) = \gcd(b,\, a \bmod b) \), with \( \gcd(a,0)=|a| \).
For multiple integers: \( \gcd(a_1,a_2,\dots,a_n) = \gcd(\gcd(a_1,a_2),a_3,\dots,a_n) \).
Alternative (prime factorization): if \( a=\prod p_i^{\alpha_i} \) and \( b=\prod p_i^{\beta_i} \), then \( \gcd(a,b)=\prod p_i^{\min(\alpha_i,\beta_i)} \).

Glossary of Variables

    - Numbers to analyze: The list of integers for which the GCF is computed (at least two; negatives and zero allowed).
    - GCF (Greatest Common Factor): The largest positive integer that divides each input number without remainder.
    - Euclidean steps: The sequence of divisions used by the Euclidean algorithm during computation.
    - Prime factorization: Representation of a number as a product of primes raised to powers.
    - Divisors of the GCF: All positive integers that divide the GCF.

How It Works: A Step-by-Step Example

Example inputs: 84, 126, 210

  1. Compute gcd(84,126): 126 = 84×1 + 42; 84 = 42×2 + 0 ⇒ gcd(84,126) = 42.
  2. Extend to the third number: gcd(42,210): 210 = 42×5 + 0 ⇒ gcd(42,210) = 42.
  3. Final result: GCF(84, 126, 210) = 42.

Prime factorization check: 84 = 2²·3·7, 126 = 2·3²·7, 210 = 2·3·5·7. Taking the minimum exponent of each common prime gives 2¹·3¹·7¹ = 42.

Frequently Asked Questions (FAQ)

What is the difference between GCF and GCD?

They are two names for the same concept: the greatest common factor/divisor of a set of integers.

Can I include zero or negative numbers?

Yes. The algorithm uses absolute values. By convention, gcd(0, n) = |n| and gcd(0, 0) = 0.

Why are decimals not allowed?

The GCF is defined for integers. If you have decimals, multiply by a power of 10 to convert to integers or round appropriately.

How many values can I enter?

Up to 50 values for performance and clarity. If you need more, consider grouping or computing iteratively.

How fast and accurate is this calculator?

It relies on the Euclidean algorithm, which is optimal and exact for integers, ensuring speed and correctness even for large values.

What are common use cases?

Simplifying fractions, finding common denominators, optimizing resource batches in manufacturing, and number theory exercises.

How does this tool compare to others?

It emphasizes accessibility (WCAG 2.1 AA), mobile-first performance, clear steps, and verifiable methodology—raising the bar on trust and usability.

Tool developed by Ugo Candido. Content verified by CalcDomain Editorial Board.
Last reviewed for accuracy on: .