Least Common Multiple

Intuition and core definition

The least common multiple (LCM) of positive integers is the smallest positive number that is a multiple of each. For $6$ and $8$, common multiples begin $24,48,72,\ldots$, so the LCM is $24$. It describes the first positive point at which repeating counts or schedules coincide.

Notation, language, and conditions

$\operatorname{lcm}(a,b)$ denotes the LCM. Using prime factorizations, include every prime appearing in any input and take the largest exponent needed. For positive $a,b$, $\gcd(a,b)\operatorname{lcm}(a,b)=ab$, which is a useful check.

Why this idea matters

The least common multiple finds the first positive total or time that several integer cycles can share exactly.

A dependable method

1. Prime-factor every input or list multiples when numbers are small.
2. Collect each distinct prime that appears.
3. For each prime, choose the greatest exponent among the inputs.
4. Multiply the selected powers.
5. Verify that every input divides the result and that the GCF–LCM product identity holds for two inputs.

Worked example

Common mistakes