GCD & LCM Calculator
Greatest common divisor and least common multiple, with steps you can follow.
GCD & LCM Calculator
Calculate Greatest Common Divisor (GCD) and Least Common Multiple (LCM) with detailed explanations
Quick Examples
GCD vs LCM
GCD
Largest positive integer that divides all inputs with no remainder. GCD(12, 18) = 6.
- GCD(a, 0) = |a|
- If GCD(a, b) = 1, a and b are coprime
- GCD(a, b) × LCM(a, b) = |a × b| for two numbers
LCM
Smallest positive integer that every input divides into evenly. LCM(12, 18) = 36.
- LCM(a, b) = |a × b| / GCD(a, b)
- If a divides b, LCM(a, b) = |b|
- Useful for fraction denominators and repeating schedules
How to compute them
Euclidean algorithm (GCD): replace the larger number with the remainder of division until the remainder is 0. The last non-zero remainder is the GCD. Example: GCD(48, 18) → 48 = 2×18 + 12 → 18 = 1×12 + 6 → 12 = 2×6 + 0 → GCD = 6.
Prime factors: for GCD take the lowest power of each shared prime; for LCM take the highest power of each prime that appears. Fast enough for small numbers; Euclidean wins for large ones.
More than two numbers: GCD(a, b, c) = GCD(GCD(a, b), c). Same pattern for LCM.
Frequently Asked Questions
What's the difference between GCD and LCM?
GCD is the biggest shared divisor. LCM is the smallest shared multiple. GCD is at most the smallest input; LCM is at least the largest.
Why is GCD × LCM = a × b?
For two numbers, the shared primes appear once in the GCD and at their highest powers in the LCM. Together they match the primes in a × b.
Euclidean or prime factorization?
Euclidean for speed on large integers. Factorization when you want to see the primes or teach the structure.
What about negative numbers?
Work with absolute values. Results are reported as positive by convention.
What does GCD = 1 mean?
The numbers are coprime (no shared factor other than 1). Then LCM(a, b) = |a × b|.
Any shortcuts?
Consecutive integers are always coprime. If one divides the other, GCD is the smaller and LCM is the larger.
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