# Definition:Lowest Common Multiple/Integers

## Definition

For all $a, b \in \Z: a b \ne 0$, there exists a smallest $m \in \Z: m > 0$ such that $a \divides m$ and $b \divides m$.

This $m$ is called the **lowest common multiple of $a$ and $b$**, and denoted $\lcm \set {a, b}$.

Note that unlike the GCD, where *either* of $a$ or $b$ must be non-zero, for the LCM *both* $a$ and $b$ must be non-zero, which is why the stipulation $a b \ne 0$.

### General Definition

This definition can be extended to any (finite) number of integers.

Let $S = \set {a_1, a_2, \ldots, a_n} \subseteq \Z$ such that $\displaystyle \prod_{a \mathop \in S} a = 0$ (that is, all elements of $S$ are non-zero).

Then the **lowest common multiple** of $S$:

- $\lcm \paren S = \lcm \set {a_1, a_2, \ldots, a_n}$

is defined as the smallest $m \in \Z_{>0}$ such that:

- $\forall x \in S: x \divides m$

## Also known as

The **lowest common multiple** is also known as the **least common multiple**.

It is usually abbreviated **LCM**, **lcm** or **l.c.m.**

The notation $\lcm \set {a, b}$ can be found written as $\sqbrk {a, b}$.

This usage is not recommended as it can cause confusion.

## Examples

### $6$ and $15$

The lowest common multiple of $6$ and $15$ is:

- $\lcm \set {6, 15} = 30$

### $25$ and $30$

The lowest common multiple of $25$ and $30$ is:

- $\lcm \set {25, 30} = 150$

### $42$ and $49$

The lowest common multiple of $42$ and $49$ is:

- $\lcm \set {42, 49} = 294$

### $27$ and $81$

The lowest common multiple of $27$ and $81$ is:

- $\lcm \set {27, 81} = 81$

### $28$ and $29$

The lowest common multiple of $28$ and $29$ is:

- $\lcm \set {28, 29} = 812$

### $n$ and $n + 1$

The lowest common multiple of $n$ and $n + 1$ is:

- $\lcm \set {n, n + 1} = n \paren {n + 1}$

### $2 n - 1$ and $2 n + 1$

The lowest common multiple of $2 n - 1$ and $2 n + 1$ is:

- $\lcm \set {2 n - 1, 2 n + 1} = 4 n^2 - 1$

## Also see

- Results about
**Lowest Common Multiple**can be found here.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 6$: The division process in $I$ - 1971: George E. Andrews:
*Number Theory*... (previous) ... (next): $\text {2-2}$ Divisibility: Exercise $4$ - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Properties of the Natural Numbers: $\S 23 \gamma$ - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 7$: Example $7.8$