MMC e MDC — Calculadora Online Grátis
Least Common Multiple and Greatest Common Factor with step-by-step solution
The LCM & GCF Calculator finds the Least Common Multiple and Greatest Common Factor of up to several integers at once, showing the full Euclidean algorithm steps for GCF and the prime factorization for LCM. These two values are building blocks for fraction arithmetic, simplification, and scheduling problems. Students working with fractions and number theory use this tool constantly.
Frequently Asked Questions
Sobre Esta Calculadora
The Greatest Common Factor (also called GCD — Greatest Common Divisor) is the largest integer that divides both numbers without a remainder. The Least Common Multiple is the smallest integer divisible by both numbers. These two quantities are mathematically linked by the identity LCM(a,b) × GCF(a,b) = a × b.
The Euclidean algorithm is one of the oldest algorithms in mathematics (described by Euclid around 300 BCE) and computes the GCF efficiently by repeated division. Once the GCF is known, the LCM follows directly. This calculator shows each step of both methods.
In practice, GCF is used to simplify fractions to lowest terms and reduce ratios. LCM is used to find common denominators when adding fractions, to synchronize periodic events (e.g., two traffic lights with cycle lengths 48s and 36s sync every 144s), and in modular arithmetic problems.
Não tenha mais dificuldade em somar frações. Esta calculadora ajuda e simplifica os seus menores divisores entre os montantes de dados.
Fórmula
GCF via Euclidean Algorithm: GCF(a,b) = GCF(b, a mod b) LCM(a,b) = |a × b| / GCF(a,b)
Exemplo Resolvido
Finding GCF and LCM of 48 and 36
Entradas:
- First number: 48
- Second number: 36
Passo a Passo:
- GCF using Euclidean algorithm:
- GCF(48, 36): 48 = 1×36 + 12 → GCF(36, 12)
- GCF(36, 12): 36 = 3×12 + 0 → GCF = 12
- LCM using formula: LCM(a,b) = |a×b| / GCF(a,b)
- LCM(48, 36) = (48 × 36) / 12 = 1728 / 12 = 144
- Verification via prime factors: 48 = 2⁴×3, 36 = 2²×3²
- LCM takes highest powers: 2⁴ × 3² = 16 × 9 = 144 ✓