Definition:Lowest Common Multiple
Contents |
Definition
Integers
For all $a, b \in \Z: a b \ne 0$, there exists a smallest $m \in \Z: m > 0$ such that $a \backslash m$ and $b \backslash m$.
This is proved in Existence of Lowest Common Multiple.
This $m$ is called the lowest common multiple (LCM) of $a$ and $b$, and denoted $\operatorname{lcm} \left\{{a, b}\right\}$.
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 $a b \ne 0$.
Integral Domain
Let $A$ be an integral domain and let $a, b \in A$ be nonzero.
Then $\ell$ is the lowest common multiple of $a$ and $b$ if both $a$ and $b$ divide $\ell$, and if $m$ is another element such that $a$ and $b$ divide $m$, then $\ell$ divides $m$.
Note
Alternatively, $\operatorname{lcm} \left\{{a, b}\right\}$ can be found written as $\left [{a, b} \right]$.
This usage is not recommended as it can cause confusion.
It is also known as the least common multiple.
Also see
- Results about Lowest Common Multiple can be found here.
Sources
- George E. Andrews: Number Theory (1971): $\S 2.2$: Exercise $4$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 23 \gamma$
