Math · Number Theory
GCF
Calculator
Find the Greatest Common Factor of up to 10 numbers. Uses prime factorization, Euclidean algorithm, and factor listing — with step-by-step solutions and fraction simplification.
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GCF
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LCM
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Coprime?
Enter Numbers
Enter 2–10 positive integers to find their Greatest Common Factor.
Quick Examples
Greatest Common Factor
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GCF
LCM
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GCF × LCM
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GCF
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LCM
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GCF × LCM
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Coprime?
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🟢 All Factors — Common Ones Highlighted
Common Factors of All Numbers
🔢 Prime Factorization — Shared Factors Highlighted
GCF = Product of shared primes (min powers)
⚙️ Euclidean Algorithm
Repeatedly divide and take remainder until remainder = 0. The last non-zero remainder is the GCF.
📋 Step-by-Step (Prime Factorization Method)
🍕 Simplify Fractions Using GCF
Using these numbers as numerator(s) and denominator — divide both by GCF to simplify.
What Is the Greatest Common Factor (GCF)?
The Greatest Common Factor (GCF) — also called Greatest Common Divisor (GCD) or Highest Common Factor (HCF) — is the largest integer that divides all given numbers without a remainder. GCF is essential for simplifying fractions, factoring polynomials, and solving ratio problems.
Three Methods to Find GCF
Method 1 — Prime Factorization:
Factor each number. Take only SHARED primes at LOWEST power.
GCF(12,18): 12=2²×3, 18=2×3² → shared: 2¹ and 3¹ → GCF = 6
Method 2 — Euclidean Algorithm:
GCF(a,b) = GCF(b, a mod b) until remainder = 0
GCF(48,18): 48=2×18+12 → GCF(18,12): 18=1×12+6 → GCF(12,6): 12=2×6+0 → GCF=6
Method 3 — Listing Factors:
Factors of 12: 1,2,3,4,6,12
Factors of 18: 1,2,3,6,9,18 → Largest common = 6
How do you simplify a fraction using GCF?
Divide both the numerator and denominator by their GCF. For example, 18/24 — GCF(18,24) = 6. Simplified: (18÷6)/(24÷6) = 3/4. This gives the simplest form because GCF is the largest number that divides both. Dividing by any smaller number won't fully reduce the fraction.
What does it mean if GCF = 1?
If GCF(a,b) = 1, the numbers are coprime (or "relatively prime") — they share no common factor other than 1. This means no fraction with these as numerator and denominator can be simplified. Consecutive integers are always coprime (e.g., 8 and 9). Prime numbers are coprime with any number they don't divide.
Why is the Euclidean algorithm efficient?
Listing factors or prime factorizing becomes slow for large numbers. The Euclidean algorithm needs only a few division steps regardless of how large the numbers are. For two numbers each with d digits, it takes at most O(d) steps. It was described by Euclid around 300 BCE and remains one of the oldest algorithms still in common use today.
What is the relationship between GCF and LCM?
For two positive integers a and b: GCF(a,b) × LCM(a,b) = a × b. This elegant relationship means you can find LCM from GCF: LCM = (a×b)/GCF. For example, GCF(12,18)=6, so LCM = (12×18)/6 = 216/6 = 36. Together, GCF and LCM completely describe the divisibility relationship between two numbers.