Definition:Lowest Common Multiple/Integers

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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