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Greatest Common Factor Calculator

Find the highest shared factor of numbers.

By Jeff Beem

Updated

Numbers

Enter two or more positive integers separated by commas. The greatest common factor (GCF) is the largest positive integer that divides all of them.

Positive integers only. Non-numeric characters are ignored. Zero and negatives are invalid.

Result
Greatest Common Factor (GCF)8

Also called Greatest Common Divisor (GCD).

Show your work

Each number as a product of primes. The GCF is the product of the lowest power of each prime common to all.

  • 16=2^4
  • 88=2^3 × 11
  • 104=2^3 × 13

Common primes (min exponent):2^3 = 8

When to Use Each Method

Use the calculator above for two or more positive integers. Formulas and definitions are in the article below.

Workflow Tips

Two numbers

Enter a pair (e.g. 48, 18). Toggle Show your work to see prime factorization or Euclidean remainder steps.

Three or more numbers

Enter comma-separated integers. The tool applies GCF pairwise; the article below explains GCF(a, b, c) = GCF(a, GCF(b, c)).

Prime factorization view

Use when you want to see shared prime "building blocks" per number. Best for smaller integers and teaching.

Euclidean view

Use when you want remainder-by-remainder steps. Best for large integers; same GCF result as the prime method.

Greatest Common Factor Calculator: GCF & GCD Solver

Free greatest common factor calculator: find GCF of two or more numbers. GCF of 3 numbers, Euclidean algorithm steps, prime factorization for GCF. Greatest common divisor solver.

What This Calculator Does & Who It's For

Calculator Purpose & Ideal Users

This greatest common factor calculator finds the GCF (or GCD) of two or more positive integers and shows your work using either the prime factorization method or the Euclidean algorithm.
  • What You'll Get:
    GCF result: The largest positive integer that divides all your numbers. Show your work: Toggle prime factorization (each number as primes, common primes to get GCF) or Euclidean algorithm (remainder steps until zero). Validation: Positive integers only; comma-separated; zero and non-numeric ignored.
  • Ideal Users:
    Students & teachers: GCF of 3 numbers calculator, how to find GCF using Euclidean algorithm, prime factorization method for GCF, find GCF of two numbers, homework and exams. Anyone: Greatest common divisor solver for simplifying fractions or divisibility. Free online, no sign-up.
  • Scope & Limits:
    Positive integers only (up to JavaScript safe integer range). No decimals or fractions. GCF and GCD are the same; this tool uses both terms.

How the Math Works

Two standard methods find the Greatest Common Factor. The prime factorization method decomposes each number into primes and multiplies shared primes at their lowest exponents: GCF(a,b)=∏pi min⁡(ea,i, eb,i)\text{GCF}(a, b) = \prod p_i^{\,\min(e_{a,i},\, e_{b,i})}Worked example: GCF(24, 36) → 24 = 2³ × 3¹, 36 = 2² × 3² → shared primes at minimum exponents give 2² × 3¹ = 12. The Euclidean algorithm uses repeated division instead: GCF(a, b) = GCF(b, a mod b), stopping when the remainder is zero. For GCF(48, 18): 48 = 18 × 2 + 12, then 18 = 12 × 1 + 6, then 12 = 6 × 2 + 0, so GCF = 6. For three or more numbers, the calculator chains pairwise results: GCF(a, b, c) = GCF(GCF(a, b), c). Both methods always produce the same answer; prime factorization is visual and intuitive for smaller numbers, while the Euclidean algorithm is faster for large integers.

How to Use This Calculator

Type two or more positive integers separated by commas into the input field, for example, 48, 18 or 16, 88, 104. The calculator displays the GCF immediately. To see how the answer was found, expand the Show your work panel and toggle between two methods: Prime factorization lists each number's prime decomposition and highlights common primes at their minimum exponents, while the Euclidean algorithm shows repeated-division remainder steps until the remainder reaches zero. For three or more numbers, the pairwise chaining is displayed step by step. Only positive integers are accepted; zero and non-numeric entries are ignored. Results stay within JavaScript's safe integer range, which covers all common homework and everyday use cases comfortably.

What Is the Greatest Common Factor (GCF)?

The greatest common factor (GCF), or greatest common divisor (GCD), of two or more integers is the largest positive integer that divides each of them with no remainder. For example, GCF(12, 18) = 6 because 6 divides both 12 and 18 and no larger integer does. GCF is used to simplify fractions (divide numerator and denominator by the GCF), factor polynomials, and solve number-theory problems. This greatest common factor calculator supports two or more numbers and shows both the prime factorization and Euclidean algorithm methods; see the FAQ above for "Is GCF the same as GCD?" and "Can you find GCF for three or more numbers?" For the full fraction workflow (operations, mixed numbers, and decimal conversion), use the fraction calculator after you know the divisor you need.

Prime Factorization Method for GCF

In the prime factorization method for GCF, write each number as a product of primes (e.g. 24 = 2³ × 3, 36 = 2² × 3²). The GCF is the product of each prime raised to the minimum exponent it appears with in any of the numbers. For 24 and 36: 2 has minimum exponent 2, 3 has minimum exponent 1, so GCF = 2² × 3¹ = 12. This method is ideal for smaller numbers and for seeing shared "building blocks." In the calculator, toggle "Prime factorization method" in Show your work to see the breakdown and common primes.

Euclidean Algorithm: How to Find GCF of Large Numbers

The Euclidean algorithm finds the GCF of two numbers by repeated division: GCF(a, b) = GCF(b, a mod b) until the remainder is 0; the GCF is then the last non-zero remainder. Example: GCF(48, 18), 48 = 18 × 2 + 12, 18 = 12 × 1 + 6, 12 = 6 × 2 + 0, so GCF = 6. For GCF of three or more numbers, apply it in sequence: GCF(a, b, c) = GCF(a, GCF(b, c)). This method is very efficient for large integers. Toggle "Euclidean algorithm" in the calculator to see a step-by-step trace; for the definition and formula, see the FAQ above.

Greatest Common Factor Calculator FAQ

Is GCF the same as GCD?

Yes. Greatest Common Factor (GCF) and Greatest Common Divisor (GCD) mean the same thing: the largest positive integer that divides all given numbers with no remainder. This calculator finds that value and shows your work using either prime factorization or the Euclidean algorithm.

What is the Euclidean algorithm?

The Euclidean algorithm finds the GCF of two numbers by repeated division: GCF(a, b) = GCF(b, a mod b) until the remainder is 0; then the GCF is the last non-zero remainder. It is very efficient even for large integers. Toggle "Euclidean algorithm" in the Show your work section to see each step.

Can you find GCF for three or more numbers?

Yes. Enter all numbers separated by commas (e.g. 16, 88, 104). The calculator uses GCF(a, b, c) = GCF(a, GCF(b, c)): it finds the GCF of the first two, then the GCF of that result with the next number, and so on. You can enter as many positive integers as you need.

How does the prime factorization method work for GCF?

Write each number as a product of primes (e.g. 16 = 2⁴, 88 = 2³ × 11). The GCF is the product of each prime raised to the smallest exponent it has in any of the numbers. For 16, 88, 104: common primes are 2 (min exponent 3), so GCF = 2³ = 8. Toggle "Prime factorization method" in Show your work to see the breakdown.

Why only positive integers?

GCF is defined for positive integers. Zero is not used (division by zero is undefined), and negative numbers are usually handled by taking absolute values; this calculator accepts only positive integers to keep the result and the steps clear for students and everyday use.

How is GCF used to simplify fractions?

Divide the numerator and denominator by their GCF to get lowest terms. Example: 18/24 → GCF(18, 24) = 6 → 3/4. After you confirm the GCF here, enter the fraction in the fraction calculator Simplifier tab to see the same reduction with a pie chart and optional mixed-number form.

Mathematical Reference Note

Calculation Logic: This tool uses standard mathematical algorithms. While we strive for accuracy, errors in logic or user input can result in incorrect data.

Verification: Results should be cross-checked if used for important academic, professional, or personal calculations.

Standard Terms: This tool is provided free of charge and as-is. CalcRegistry provides no warranty regarding the accuracy or fitness of these results for your specific needs.

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