# Definition:Lowest Common Multiple/Integers

## Definition

### Definition 1

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 $\lcm \set {a, b}$.

### Definition 2

Let $a, b \in \Z$ be integers such that $a b \ne 0$.

Then the **lowest common multiple** of $a$ and $b$ is the (strictly) positive integer $m$ which satisfies:

- $(1): \quad a \divides m$ and $b \divides m$
- $(2): \quad $If there exists $c \in \Z_{>0}$ such that $a \divides c$ and $b \divides c$, then $m \le c$

where $\divides$ denotes divisibility.

### 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 $\ds \prod_{a \mathop \in S} a = 0$ (that is, all elements of $S$ are non-zero).

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

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

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

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

## Warning

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.

Hence the stipulation:

- $a b \ne 0$

## 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:

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

### $-12$ and $30$

The lowest common multiple of $-12$ and $30$ is:

- $\lcm \set {-12, 30} = 60$

### $25$ 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**.