What are GCD and LCM?

GCD (greatest common divisor, also called HCF) is the largest number dividing all inputs evenly; LCM (least common multiple) is the smallest number all inputs divide into. For 12 and 18: GCD = 6, LCM = 36.

How does the Euclidean algorithm find GCD?

Repeatedly replace the larger number with the remainder of dividing it by the smaller, until remainder is 0 — the last nonzero remainder is the GCD. GCD(48,18): 48 mod 18=12, 18 mod 12=6, 12 mod 6=0 → GCD 6. The calculator displays each step.

How are GCD and LCM related?

For two numbers: GCD × LCM = a × b. So LCM(12,18) = 12×18 ÷ 6 = 36 — which is how the calculator computes LCM without listing multiples.

Can it handle more than two numbers?

Yes — GCD and LCM extend by folding: GCD(a,b,c) = GCD(GCD(a,b),c), and similarly for LCM. Enter any list.

What are GCD and LCM actually used for?

GCD simplifies fractions to lowest terms and splits things into equal groups; LCM finds common denominators, synchronizes repeating cycles (buses arriving every 12 and 18 minutes align every 36), and schedules recurring events.

GCD & LCM Calculator — Greatest Common Divisor & LCM...

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GCD & LCM Calculator — With the Working Shown

Simplifying 48/60, adding fractions with denominators 8 and 12, or figuring out when two repeating schedules coincide — GCD and LCM are the tools. This free calculator finds both for any list of numbers, showing the Euclidean algorithm step-by-step and the prime-factorization method side by side, so the answer teaches the technique. In-browser, no signup.

Two Methods, One Answer

Euclidean algorithm (fast, elegant): GCD(270, 192) → 270 mod 192 = 78 → 192 mod 78 = 36 → 78 mod 36 = 6 → 36 mod 6 = 0. GCD = 6, in four steps regardless of number size — the same algorithm inside every computer algebra system.

Prime factorization (visual): 270 = 2·3³·5 and 192 = 2⁶·3. GCD takes the lowest shared powers (2¹·3¹ = 6); LCM takes the highest of every prime (2⁶·3³·5 = 8640). Seeing both builds the intuition the algorithm hides.

The Identity That Connects Them

GCD(a, b) × LCM(a, b) = a × b
6 × 8640 = 270 × 192 = 51,840 ✓

One consequence: coprime numbers (GCD 1) have LCM equal to their product — why denominators like 7 and 9 need denominator 63, no shortcut available.

Real Problems These Solve

Fractions — divide numerator and denominator by their GCD: 48/60 → 4/5. Common denominators are LCMs.
Scheduling — events every 6, 10, and 15 days coincide every LCM(6,10,15) = 30 days.
Manufacturing and packing — largest equal batches from mixed quantities is a GCD question; gear-teeth alignment cycles are LCM.
Programming — array rotations, cryptographic key checks (RSA requires GCD(e, φ)=1), and rhythm generation in music code.

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Any quantity of numbers, full working shown, computed locally — homework help that actually explains.

Gcd Lcm Calculator

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