LCM & HCF/GCF Calculator

How It Works

All three tabs compute the same answers with different machinery, and the Steps panel reruns the chosen method on your own numbers every time you type. For a preview, take 12 and 18: 12 = 2²×3 and 18 = 2×3². The GCF takes each prime to its lowest power (2¹×3¹ = 6), the LCM takes each prime to its highest (2²×3² = 36), and Euclid reaches 6 by remainders: 18 = 1×12 + 6, then 12 = 2×6 + 0. Different routes, one answer each.

The factor min/max principle

Common divisors can only use primes that appear in every number, and only up to the smallest exponent — so GCF = product of minimum powers. Common multiples must contain every prime from every number at least up to the largest exponent — so LCM = product of maximum powers. With 12 and 18 that is min(2,1) = 1 for the 2s, min(1,2) = 1 for the 3s, and the maxes 2 and 2. This generalizes to any count of inputs.

Euclid, with remainders instead of factoring

gcd(a, b) = gcd(b, a mod b): any number dividing both a and b also divides their difference-and-remainder, so the pair shrinks without losing common divisors. For 12 and 18 the chain 18, 12 → 12, 6 → 6, 0 stops at 6. For more than two numbers the tool chains the pairs, and gcd(0, n) = n makes chains start cleanly even with zeros.

Why LCM is verified by division

After computing, the panel checks LCM ÷ each input leaves remainder 0 — the defining property of a common multiple — and the smallest such positive number is what the maximum-exponent construction guarantees. If the inputs include 0, the verification prints the convention note instead, because 0 is technically a multiple of every integer.

Frequently Asked Questions