How do I use the Greatest Common Factor Calculator?

Enter the required values, then calculate. The Greatest Common Factor Calculator shows results, formulas, and step-by-step work when available. No account is required.

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How it works

The greatest common factor is the largest positive integer that divides every input. Methods include Euclidean gcd(a, b) = gcd(b, a mod b) and multiplying lowest common prime powers.

gcd(a, b) = gcd(b, a mod b)

When to use

Use this tool when you need to: try 330, 75, 450, 225 (GCF = 15) or 16, 88, 104 (GCF = 8). Related: LCM and Factor calculators.

How to use this tool

Enter your values in the fields above.
Review the formula and any mode options for your problem.
Read the result and the step-by-step solution.

Try 330, 75, 450, 225 (GCF = 15) or 16, 88, 104 (GCF = 8). Related: LCM and Factor calculators.

Frequently asked questions

What is the Greatest Common Factor Calculator?: The greatest common factor is the largest positive integer that divides every input. Methods include Euclidean gcd(a, b) = gcd(b, a mod b) and multiplying lowest common prime powers.
How do I use the Greatest Common Factor Calculator?: Enter the required values, then calculate. The Greatest Common Factor Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Greatest Common Factor Calculator use?: gcd(a, b) = gcd(b, a mod b)

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Greatest common factor

Enter two or more non-zero integers separated by commas. The GCF (also called GCD) is the largest positive integer that divides every number with no remainder.

How it works

What is the greatest common factor (GCF)?

In mathematics, the greatest common factor (GCF) — also known as the greatest common divisor (GCD) — of two or more non-zero integers is the largest positive integer by which each of them can be divided. It is commonly written GCF(a, b) or gcd(a, b). Example: GCF(32, 256) = 32, because 32 divides both numbers and no larger positive integer does. Another example: the positive divisors of 48 include 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Those of 18 include 1, 2, 3, 6, 9, 18. The largest shared value is 6, so GCF(48, 18) = 6.

Prime factorization method

Factor each integer into primes, keep only primes that appear in every factorization, and take the lowest power of each shared prime. Multiply those powers to get the GCF. Example: GCF(16, 88, 104) 16 = 2 × 2 × 2 × 2 = 2⁴ 88 = 2 × 2 × 2 × 11 = 2³ × 11 104 = 2 × 2 × 2 × 13 = 2³ × 13 Common primes (lowest powers): 2³ = 8. So GCF(16, 88, 104) = 8. Prime factorization is practical for small values; large integers make factoring by hand tedious.

Euclidean algorithm

A faster method uses the Euclidean algorithm: the GCD of two integers also divides their difference (and more usefully their remainder under division). In modern form: 1. Write a = b·q + r with remainder r = a mod b. 2. Replace (a, b) with (b, r). 3. Repeat until the remainder is 0; the last non-zero remainder is the GCD. Example: GCF(268442, 178296) reduces by successive remainder division and equals 2. For more than two integers, reduce pairwise: after GCF(a, b) = q, compute GCF(q, c), and so on. Example: if GCF(268442, 178296) = 2, then GCF(268442, 178296, 66888) = GCF(66888, 2) = 2. This calculator uses the Euclidean algorithm for the official result and also shows the prime-factorization path when it clarifies shared primes.

Related calculators

LCM finds the least shared multiple; factor and prime tools inspect the individual integers; common factors lists every shared divisor, not only the greatest.

How it works

The greatest common factor is the largest positive integer that divides every input. Methods include Euclidean gcd(a, b) = gcd(b, a mod b) and multiplying lowest common prime powers.

gcd(a, b) = gcd(b, a mod b)

When to use

Use this tool when you need to: try 330, 75, 450, 225 (GCF = 15) or 16, 88, 104 (GCF = 8). Related: LCM and Factor calculators.

How to use this tool

Enter your values in the fields above.
Review the formula and any mode options for your problem.
Read the result and the step-by-step solution.

Try 330, 75, 450, 225 (GCF = 15) or 16, 88, 104 (GCF = 8). Related: LCM and Factor calculators.

Frequently asked questions

What is the Greatest Common Factor Calculator?: The greatest common factor is the largest positive integer that divides every input. Methods include Euclidean gcd(a, b) = gcd(b, a mod b) and multiplying lowest common prime powers.
How do I use the Greatest Common Factor Calculator?: Enter the required values, then calculate. The Greatest Common Factor Calculator shows results, formulas, and step-by-step work when available. No account is required.
What formula does the Greatest Common Factor Calculator use?: gcd(a, b) = gcd(b, a mod b)