Calculate the Greatest Common Divisor and Least Common Multiple of up to 10 integers.
The Greatest Common Divisor (GCD) of a set of integers is the largest number that divides all of them with no remainder; the Least Common Multiple (LCM) is the smallest positive integer that all of them divide into. The tool finds the GCD with the iterative Euclidean algorithm, repeatedly replacing the larger number with the remainder of dividing it by the smaller, and derives the LCM from the identity LCM(a, b) = (a ÷ GCD(a, b)) · b, extending both to as many as 10 numbers by folding pairwise: GCD(a, b, c) = GCD(GCD(a, b), c). It also shows the prime factorization of each input by successive division, the classic by-hand method. All arithmetic uses BigInt, so inputs up to 10³⁰ stay exact; an LCM that would exceed 10³⁰ is flagged rather than printed. The fundamental identity a · b = GCD(a, b) · LCM(a, b) ties the two concepts together.
The GCD is computed with the Euclidean algorithm: divide the larger by the smaller, keep the remainder, and repeat with (smaller, remainder) until the remainder is zero, the last non-zero divisor is the GCD. For more than two numbers, apply it in cascade, pair by pair.
The LCM follows from the identity GCD(a, b) · LCM(a, b) = a · b, so LCM(a, b) = (a ÷ GCD(a, b)) · b. Prime factorization confirms the result: the GCD takes the LOWEST power of each shared prime, and the LCM takes the HIGHEST power of every prime that appears.
Example 1, two numbers: 12 = 2² · 3 and 18 = 2 · 3². The GCD takes 2¹ · 3¹ = 6; the LCM takes 2² · 3² = 36. Checking with the identity: 12 · 18 = 216 = 6 · 36.
Example 2, three numbers (8, 12, 20): GCD(8, 12) = 4 and GCD(4, 20) = 4, so GCD = 4. For the LCM: LCM(8, 12) = 24 and LCM(24, 20) = 120, so LCM = 120 (the smallest number divisible by 8, 12, and 20).
When the numbers are coprime (no common factor other than 1), the GCD is 1 and the LCM is simply the product of all of them. Zero is not accepted, because every number divides zero (leaving the GCD undefined) and the LCM would be meaningless, by convention, GCD and LCM are defined for positive integers.
The GCD is used to reduce fractions to lowest terms and to split quantities into equal groups; the LCM is used to find the lowest common denominator, to synchronize periodic events (traffic lights, gears), and to solve problems about repeating cycles.
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<iframe
src="https://jkit.tools/embed/en-US/gcd-lcm-calculator"
width="100%"
height="600"
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title="GCD & LCM Calculator"
></iframe>These references help contextualize formulas, standards, APIs and limitations used on this page. They do not replace professional validation when a result has legal, financial, medical or operational impact.
GCD (Greatest Common Divisor) is the largest positive integer that divides all numbers in the set without a remainder. It can be found with the Euclidean algorithm or by comparing prime factorizations and taking the lowest power of each shared prime. For example, GCD(12, 18) = 6.
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