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:
- $\lcm \set {28, 29} = 812$
$3$, $9$ and $11$
The lowest common multiple of $3$, $9$ and $11$ is:
- $\lcm \set {3, 9, 11} = 99$
$7$, $9$, $12$ and $14$
The lowest common multiple of $7$, $9$, $12$ and $14$ is:
- $\lcm \set {7, 9, 12, 14} = 252$
$n$ 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 the lowest common multiple can be found here.