Calculadora de MDC e MMC

The greatest common factor (GCF), also called the greatest common divisor (GCD), is the largest whole number that divides both numbers evenly. This calculator finds it with the Euclidean algorithm: repeatedly divide and keep the remainder until it reaches zero.

The least common multiple (LCM) is the smallest number that both inputs divide into evenly. It relates directly to the GCF: LCM(a, b) = (a × b) ÷ GCF(a, b). LCM is especially useful for finding a common denominator when adding fractions.

Example: for 12 and 18, the GCF is 6 (the largest number that divides both) and the LCM is 36 (12 × 18 ÷ 6). That means 12/18 simplifies to 2/3, and 36 is the smallest number both 12 and 18 divide into evenly.

The Euclidean algorithm behind GCF is fast even for large numbers — divide the larger by the smaller, replace the larger with the remainder, repeat until remainder is zero.

LCM is the smallest number both inputs divide evenly — useful for syncing repeating events (bus every 4 days, train every 6 days meet every 12 days).

Worked example with defaults: GCF(12, 18) = 6 because 6 is the largest divisor of both. LCM = 12×18÷6 = 36 — the smallest number divisible by both 12 and 18.

Reducing 48/72 before cross-multiplying in proportion problems — GCF(48,72)=24 gives 2/3 in one step instead of dividing numerator and denominator separately after the fact. GCF(48,72)=24 reduces 48/72 to 2/3 before cross-multiplying proportions — one gcd pass beats simplifying numerator and denominator separately. Listing gcd steps beside 48/72 → 2/3 helps graders verify you reduced before cross-multiplying.

Scheduling two buses leaving every 12 and 18 minutes syncs at LCM 36 minutes — GCF/LCM pairs appear in discrete math word problems beyond fraction simplification. Buses every 12 and 18 minutes meet every LCM 36 minutes — discrete scheduling drills use the same lcm(12,18)=36 as fraction denominators. Note lcm(12,18)=36 on your timeline sketch when buses sync in discrete scheduling word problems.

Prime factorization backup for 12=2²×3 and 18=2×3² — GCF takes minimum powers (2×3=6), LCM takes maximum (2²×3²=36), matching the Euclidean algorithm output for defaults. Prime factors 12=2²×3 and 18=2×3² — GCF min powers → 6, LCM max powers → 36, matching Euclidean output for defaults twelve and eighteen.