What is the least common multiple?

The least common multiple (LCM) of two or more positive integers is the smallest positive integer that each of them divides into exactly. For example, the LCM of 4 and 6 is 12, because 12 is the smallest number that is both a multiple of 4 and a multiple of 6.

How is the LCM calculated?

The fastest method uses the GCD: LCM(a,b) = (a / GCD(a,b)) x b. Dividing by the GCD first keeps the intermediate value as small as possible and avoids overflow. For three or more numbers the calculator chains this pairwise, left to right, since LCM(a,b,c) = LCM(LCM(a,b),c).

What is the prime-factor method for finding the LCM?

Each number is factorised into prime powers. For each prime that appears in any number, take the highest exponent seen. Multiply these highest powers together to get the LCM. For example, 12 = 2^2 x 3 and 18 = 2 x 3^2 give LCM = 2^2 x 3^2 = 36. The table in this tool highlights the winning exponent in each column.

What is the difference between LCM and GCD?

The GCD (greatest common divisor) is the largest number that divides all your inputs exactly, while the LCM is the smallest number that all your inputs divide into exactly. For two numbers a and b the identity GCD(a,b) x LCM(a,b) = a x b always holds, so knowing one gives you the other.

When are two numbers coprime, and what is their LCM?

Two integers are coprime (relatively prime) when their GCD is 1 -- they share no common prime factor. In that case LCM(a,b) = a x b, their plain product, because there is no shared factor to cancel.

Why is the LCM useful in real life?

The LCM arises whenever you need to synchronise two repeating cycles. Classic examples include: adding fractions (the LCM of the denominators gives the common denominator), scheduling (when two events with different periods next coincide -- two buses with 4-minute and 6-minute headways next meet after 12 minutes), music (when two rhythmic patterns of different lengths next align), and gear ratios (when two meshed gears of different tooth counts return to the same relative position).

Is there an LCM for more than two numbers?

Yes. LCM extends to any count of integers by folding pairwise from left to right. The calculator shows every fold step and the corresponding GCD used in each one, so you can trace the exact derivation for three, four, or more numbers.

Least Common Multiple Calculator

The least common multiple (LCM), also called the lowest common multiple, of a set of positive integers is the smallest positive integer that every member of the set divides into without a remainder. It is one of the most-used concepts in arithmetic: you need it every time you add or subtract fractions with different denominators, and it reappears in scheduling, music theory, gear-ratio analysis, and any other context where two or more repeating cycles must be synchronised.

This calculator accepts two or more positive whole numbers, computes the LCM instantly, and then walks you through three levels of explanation: a prime-factor table showing which exponent each prime contributes, a step-by-step pairwise fold so you can follow the chain of GCD divisions, and a visual cycle diagram (for two-number inputs up to 1,000) that draws tick marks where each number’s multiples fall and highlights the first position where both sequences land at the same time.

How it works

The core formula is:

LCM(a, b) = (a / GCD(a, b)) x b

Dividing by the GCD before multiplying keeps the intermediate result as small as possible, which matters for large inputs. For three or more numbers the calculator chains this left to right, because LCM(a, b, c) = LCM(LCM(a, b), c), and so on. The GCD at each step is found via the Euclidean algorithm.

Equivalently, via prime factorisation: write each number as a product of prime powers, then for every prime that appears in any of the numbers take the highest exponent seen, and multiply those powers together. For example:

4 = 2^2
6 = 2^1 x 3^1
10 = 2^1 x 5^1

The primes involved are 2, 3, and 5. The highest exponents are 2^2, 3^1, and 5^1, giving LCM = 4 x 3 x 5 = 60.

The tool’s prime-factor table makes this visual: each column is a prime, each row is one of your numbers, and the highlighted entry in each column is the maximum exponent that feeds into the LCM.

Worked example

Find the LCM of 12, 18, and 30:

Factorisations: 12 = 2^2 x 3, 18 = 2 x 3^2, 30 = 2 x 3 x 5.

Take the maximum exponent of each prime: 2^2, 3^2, 5^1.

LCM = 4 x 9 x 5 = 180.

Pairwise fold confirms: LCM(12, 18) = (12 / GCD(12,18)) x 18 = (12/6) x 18 = 2 x 18 = 36; then LCM(36, 30) = (36 / GCD(36,30)) x 30 = (36/6) x 30 = 6 x 30 = 180.

Numbers	LCM	Notes
4, 6	12	common denominator for 1/4 + 1/6
4, 6, 10	60	three-number fold
12, 18, 30	180	see example above
8, 15	120	coprime pair (GCD = 1), so LCM = product
100, 75	300	GCD = 25, so LCM = 100 x 75 / 25

Formula note. LCM(a,b) = (a / GCD(a,b)) x b, where GCD is found by the Euclidean algorithm. For any two positive integers the identity GCD(a,b) x LCM(a,b) = a x b always holds. The calculator prints this identity check for two-number inputs.

The cycle diagram draws tick marks at every multiple of each number up to the LCM, making it immediately clear that the two sequences first coincide at exactly one point within that window. All calculations run entirely in your browser; no data is uploaded or stored.