Two questions come up constantly with pairs of numbers: what's the largest number that divides both (GCF), and what's the smallest number both divide into (LCM)? They power fraction simplification, scheduling problems, and gear ratios alike.
Como funciona
The calculator finds the GCF with Euclid's algorithm — repeatedly replacing the larger number by the remainder of dividing it by the smaller, until the remainder is zero — then derives the LCM from the elegant identity linking the two.
A fórmula
Euclid's algorithm: GCF(a, b) = GCF(b, a mod b), terminating when b = 0. It's one of the oldest algorithms still in daily use, described around 300 BC.
Exemplo resolvido
For 36 and 60: 60 mod 36 = 24, 36 mod 24 = 12, 24 mod 12 = 0, so GCF = 12 — and LCM = 36 × 60 ÷ 12 = 180.
Perguntas frequentes
What's a practical use of the LCM?
Repeating cycles. If one bus comes every 36 minutes and another every 60, they coincide every LCM(36,60) = 180 minutes. The same logic schedules maintenance, meshing gears, and common denominators.
What does it mean if the GCF is 1?
The numbers are coprime — they share no factor. Fractions built from them are already in lowest terms, and their LCM is simply their product.
How does the GCF simplify fractions?
Divide numerator and denominator by their GCF in one step. 36/60 with GCF 12 becomes 3/5 immediately — no trial-and-error halving needed.