LCM Calculator — Least Common Multiple
Enter two or three positive whole numbers and instantly find their Least Common Multiple — the smallest positive integer that every entered number divides into evenly.
Smallest positive integer divisible by all entered numbers
- 1
GCD of the two numbers
GCD(12, 18) = 6Computed with the Euclidean algorithm: repeatedly replace with the remainder. - 2
LCM = a × b ÷ GCD
12 × 18 ÷ 6 = 36
此計算機如何運作？
LCM(a, b) = |a × b| / GCD(a, b). Compute GCD via the Euclidean algorithm (repeatedly replace with the remainder), then divide into the product. For three numbers chain it: LCM(LCM(a,b), c). The LCM is the smallest positive integer divisible by every input.
公式
How this is calculated
The Least Common Multiple of two integers is found most efficiently via the GCD: LCM(a, b) = |a × b| / GCD(a, b). The GCD is computed with the Euclidean algorithm — repeatedly replace the larger number with the remainder when divided by the smaller, until the remainder is zero; the last non-zero value is the GCD. This never requires factoring the numbers, making it fast even for large values.
For three numbers the computation is chained: LCM(a, b, c) = LCM(LCM(a, b), c). The order does not matter because LCM is commutative and associative.
Only positive integers are accepted. Zero would make LCM = 0 by convention (since every number divides 0), which is usually not useful. If you need the LCM for fraction work (finding a common denominator) the result here is identical to the Least Common Denominator.
常見問題
For any two positive integers a and b: LCM(a, b) × GCD(a, b) = a × b. This means LCM = a × b / GCD, and GCD = a × b / LCM. They are complementary: the GCD measures the largest common factor; the LCM measures the smallest common multiple.
LCM is used to find a common denominator when adding or subtracting fractions, to synchronise repeating events (e.g. two gears meeting at the same position), and in number theory problems involving divisibility and modular arithmetic.
No. The LCM is always at least as large as the largest of the input numbers, because every number is a multiple of itself. When one number is a multiple of all others (e.g. 6 and 3) the LCM equals the largest number.
也稱為
TG we-Calculate Editorial Team. (2026). LCM Calculator — Least Common Multiple [Online calculator]. TG we-Calculate. https://we-calculate.com/zh-tw/calculator/lcm-calculator
TG we-Calculate Editorial Team. "LCM Calculator — Least Common Multiple." TG we-Calculate. 2026. https://we-calculate.com/zh-tw/calculator/lcm-calculator.
TG we-Calculate Editorial Team, "LCM Calculator — Least Common Multiple," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/zh-tw/calculator/lcm-calculator
@misc{wecalculate_lcm_calculator, title = {LCM Calculator — Least Common Multiple}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/zh-tw/calculator/lcm-calculator}}, year = {2026}, note = {TG we-Calculate} }
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