Algebra 1 Basic

GCF & LCM Calculator

Find the Greatest Common Factor and Least Common Multiple of two or three numbers — with prime factorization steps and the Euclidean algorithm.

Prime factorization method
Euclidean algorithm
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Greatest Common Factor
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Least Common Multiple
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Step-by-Step Solution
GCF & LCM — Key Formulas
GCF(a, b) = product of shared prime factors (lowest exponent) LCM(a, b) = product of all prime factors (highest exponent) GCF(a, b) × LCM(a, b) = a × b

The Euclidean algorithm finds GCF efficiently by repeated division: GCF(a, b) = GCF(b, a mod b). Repeat until remainder = 0.

For three numbers: GCF(a, b, c) = GCF(GCF(a, b), c) and LCM(a, b, c) = LCM(LCM(a, b), c).

The Euclidean algorithm is the fastest method by hand — no factoring needed!
When to Use GCF vs LCM
  • GCF — simplifying fractions (divide numerator & denominator by GCF)
  • GCF — splitting into equal groups (largest group size)
  • GCF — factoring out the greatest common factor from an expression
  • LCM — adding or subtracting fractions (find common denominator)
  • LCM — scheduling repeating events (when do they next coincide?)
  • LCM — finding the smallest number divisible by both a and b
Remember: GCF ≤ min(a, b) and LCM ≥ max(a, b) — always check your answer!
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