Least common multiple

Definition

Let $a$ and $b$ be nonzero integers. A common multiple of $a$ and $b$ is an integer $m$ such that $a \mid m$ and $b \mid m$. The least common multiple $\operatorname{lcm}(a, b)$ is the smallest positive common multiple of $a$ and $b$. By convention, $\operatorname{lcm}(a, 0) = \operatorname{lcm}(0, a) = 0$ for each integer $a$.

Remarks

The positive integer $|a b|$ is a common multiple, so $\operatorname{lcm}(a, b)$ exists and $\operatorname{lcm}(a, b) \le |a b|$. Each common multiple of $a$ and $b$ is a multiple of $\operatorname{lcm}(a, b)$. For example, $\operatorname{lcm}(4, 6) = 12$. The product of the gcd and the lcm is the product of the two numbers.

Explanations

L1 human

List the positive multiples of each number, and take the smallest number that is in both lists. The multiples of $4$ are $4, 8, 12, 16, \dots$ and the multiples of $6$ are $6, 12, 18, \dots$. Thus $\operatorname{lcm}(4, 6) = 12$. You use this number when you add $\tfrac{1}{4} + \tfrac{1}{6}$ with a common denominator.