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GCF & LCM Calculator

Find the Greatest Common Factor (GCF) and Least Common Multiple (LCM) of multiple integers.

GCF & LCM — interactive calculator, computed instantly in your browser.

📚 Authoritative Educational Guide

GCF & LCM Guide: Prime Factorization & Euclidean Algorithm

🔍 Number Theory

Venn Diagram Relationship: GCF vs LCM for 24 and 36

2 2, 2, 3 GCF = 12 3 Number 24 ($2^3 \times 3$) Number 36 ($2^2 \times 3^2$) LCM = All Factors = $2 \times 2 \times 2 \times 3 \times 3 = \mathbf{72}$

Fundamental Duality: The GCF is the product of common prime factors (intersection), while the LCM is the product of all prime factors at highest powers (union).

The Greatest Common Factor (GCF / GCD) is the largest positive integer that divides two or more integers without leaving a remainder. The Least Common Multiple (LCM) is the smallest positive integer divisible by all given numbers.

1. The Euclidean Algorithm & GCF-LCM Connection

The Euclidean algorithm provides an extremely efficient method for finding the GCF:

\text{GCD}(a, b) = \text{GCD}(b, a \bmod b) \quad \text{until remainder is 0}

For any two integers $a$ and $b$, their product equals the product of their GCF and LCM:

a \times b = \text{GCF}(a, b) \times \text{LCM}(a, b) \implies \text{LCM}(a, b) = \frac{|a \times b|}{\text{GCF}(a, b)}

2. Real-World Applications

  • Simplifying Fractions: Divide numerator and denominator by their GCF.
  • Adding Fractions with Different Denominators: The Lowest Common Denominator (LCD) is precisely the LCM of denominators.
  • Repeating Cycles & Scheduling: Finding when two periodic events (traffic lights, planetary orbits, manufacturing maintenance schedules) synchronize.

🔗 Divisibility & Multiple Calculations

GCF and LCM calculations are fundamental to simplifying algebraic fractions and finding common denominators. Visit our ➗ Math Hub.

Examine full prime factor trees or simplify ratio comparisons using the companion solvers below.

✨Prime Factorization Tree🔢Fraction Common Denominators⚖️Ratio Simplification Tool

Frequently Asked Questions (GCF & LCM)

What if two numbers have a GCF of 1?

Numbers with a GCF of 1 are called 'coprime' or 'relatively prime' (e.g. 8 and 15). Their LCM is simply their direct product ($8 \times 15 = 120$).

Can GCF or LCM be applied to more than two numbers?

Yes! For three numbers $a, b, c$, compute $\text{GCF}(a, b, c) = \text{GCF}(\text{GCF}(a, b), c)$. The same associative property applies to LCM.