GCF and LCM are mirror twins: the greatest common factor is the largest number that divides into everything on your list, and the least common multiple is the smallest number everything divides into. This calculator computes both for up to 20 numbers, and shows the prime factorizations that explain them.
Reading the primes
48 = 2⁴·3 and 36 = 2²·3². For the GCF, take each shared prime at its lowest power: 2²·3 = 12. For the LCM, take every prime at its highest: 2⁴·3² = 144. The elegant identity a×b = GCF×LCM (48×36 = 12×144 = 1,728) ties the pair together for any two numbers.
How the computation actually runs
The Euclidean algorithm needs no factoring at all — just remainders: GCF(48, 36) → 48 mod 36 = 12 → 36 mod 12 = 0 → answer 12. It's among the oldest algorithms still in daily use (Euclid, ~300 BC) and blazes through numbers of any size, which is why this page happily takes big integers via BigInt.
What each one is for
GCF problems sound like "largest equal groups": cutting 48 cm and 36 cm boards into the longest equal pieces (12 cm), or simplifying 36⁄48 to 3⁄4 — the exact move the fraction calculator automates. LCM problems sound like "when do they align?": buses every 12 and 18 minutes meet every 36; adding 1⁄12 + 1⁄18 wants the common denominator 36.
Coprime, and other quick reads
If the GCF is 1, the numbers are coprime and the LCM is their full product — no shared structure to exploit. Consecutive integers are always coprime. For the factor lists themselves, digit by digit, the prime factorization calculator decomposes any number completely.
Exact BigInt arithmetic; LCM display capped at 10³⁰.