Greatest Common Divisor (GCD) & LCM Calculator
The problem: Manual fraction reduction and multi-number common multiple calculations are tedious and error-prone.
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Greatest Common Divisor (GCD) & LCM Calculator: the complete guide
The Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Calculator computes the highest common factor and smallest shared multiple for up to 10 numbers simultaneously using the Euclidean division algorithm and prime factorization. View step-by-step division ladders, prime factor decomposition trees, and coprime status verifications for elementary mathematics and advanced fraction reduction.
The Euclidean Algorithm — 2,300 Years of Efficiency
Documented by Greek mathematician Euclid in Book VII of his 'Elements' around 300 BC, the Euclidean algorithm remains one of the most efficient mathematical algorithms ever devised. It operates on the theorem that the greatest common divisor of two numbers does not change if the larger number is replaced by its remainder when divided by the smaller number: GCD(a, b) = GCD(b, a mod b).
By iterating this division process until the remainder reaches zero, the last non-zero remainder represents the exact GCD. Our tool logs every division equation step-by-step so students can follow the working for homework verification.
Connecting GCD and LCM via Prime Factorization
For any two positive integers a and b, their product equals the product of their greatest common divisor and least common multiple: a × b = GCD(a, b) × LCM(a, b). This fundamental relationship allows rapid derivation of the LCM once the GCD is known: LCM(a, b) = (a × b) / GCD(a, b).
When extending to three or more numbers, prime factorization reveals the structural relationship: the GCD takes the minimum exponent of each shared prime factor, while the LCM takes the maximum exponent across all prime factors present.
Step by step: how to use GCD & LCM Calc
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Enter two or more positive integers separated by commas or spaces (e.g., '24, 36, 60').
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View the calculated Greatest Common Divisor (GCD) and Least Common Multiple (LCM).
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Check the Coprime indicator (GCD = 1 indicates no shared factors greater than 1).
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Examine the Prime Factorization panel to see each number decomposed into prime exponents.
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Review the Euclidean Algorithm steps showing the exact quotient and remainder sequence.
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