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
- 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.
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 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?"
Prime Factorization Method for GCF
In the prime factorization method for GCF, write each number as a product of primes (e.g. 24 = 2ยณ ร 3, 36 = 2ยฒ ร 3ยฒ). 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 = 2ยฒ ร 3ยน = 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) โ 48 = 18 ร 2 + 12, 18 = 12 ร 1 + 6, 12 = 6 ร 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 = 2โด, 88 = 2ยณ ร 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 = 2ยณ = 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.