Least Common Multiple Calculator finds the smallest positive shared multiple of nonzero integers, and returns zero under the documented zero-input convention.
Build the least common multiple one pair at a time
For nonzero integers, the least common multiple is the smallest positive integer divisible by every input. Enter integer values separated by commas, spaces or new lines. The result is exact; it is not a rounded product or an approximation. For a nonzero pair, find the GCD, divide one magnitude by that GCD, then multiply by the other magnitude. The division removes shared factors that a direct product counts twice. For a longer list, combine the first pair, then find the LCM of that accumulated result and the next integer. The tool shows those folds and their Euclidean divisions. OpenStax’s prime-factor treatment explains why each required prime power must appear in the resulting multiple.
Keep the original signs and omit thousands separators, decimal points and scientific notation.
Review the named primary result and open calculation details to check the arithmetic.
LCM of three integers: worked example
The overlapping-factor preset enters 6, 10, 15. The LCM is 30 and the list’s GCF is 1. These are the engine’s steps:
lcm(6, 10) = 30
6 = 0 × 10 + 6
10 = 1 × 6 + 4
6 = 1 × 4 + 2
4 = 2 × 2 + 0
lcm(30, 15) = 30
30 = 2 × 15 + 0
The simple pair 24 and 36 returns LCM 72. Using the GCD once works for that pair; a longer list requires repeated pairwise calculation.
Prime powers and common denominators
The prime-factor method keeps each distinct prime at the largest exponent needed by any input. It does not multiply every repeated factor from every input. The denominator preset decomposes its entries as 27 = 3^3; 18 = 2 × 3^2. Its LCM is 54.
For positive integer fraction denominators, their LCM is the least common denominator. Each old denominator divides that answer exactly, so the numerator and denominator can be multiplied by the corresponding integer scale without changing the fraction. Enter only the integer denominators here; this page does not parse whole fractions or algebraic expressions. The optional prime explanation is a comparison aid, while the exact pairwise method remains the primary calculation.
Zero, signed values and the two-number product rule
Negative signs do not change the positive LCM of nonzero integers. If any input is zero, this tool returns 0 under the documented Python math convention. A positive multiple of zero does not exist, so that convention must be distinguished from the ordinary smallest-positive-multiple definition.
The identity GCF × LCM = absolute value of the product applies to a pair. It cannot be extended by replacing the pair product with the product of an entire list. The worked example demonstrates that the pairwise folds are necessary when factors overlap across several entries.
Enter two through 32 integers, with up to 64 digits per value and no thousands separators, decimal points or scientific notation. Optional prime explanations accept nonzero entries with up to 14 digits. All displayed integer answers remain exact strings. Larger prime explanations and malformed entries are rejected explicitly.
The published integer tools separate these related outputs: Greatest Common Factor Calculator. The primary result and required inputs identify the appropriate job.
The Prime Factorization Calculator decomposes one integer into prime powers and repeated factors; this differs from comparing a list.
A complete divisor list and product-checked factor pairs belong in the Factors Calculator, which also explains signed inputs.
When you need coefficients that reproduce the GCD, the Extended Euclidean Algorithm Calculator checks the original-input identity and its inverse residue.