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Def/Least common multiple
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Definition of Least common multiple: Suppose that $R$ is a Euclidean domain (such as $\ZZ$ or $\RR[X]$, for example). Suppose that $a$ and $b$ are elements of $R$, and $a$ or $b$ is nonzero. A least common multiple of $a$ and $b$, sometimes called $LCM(a,b)$ (especially when $R = \ZZ$), is an element $\ell \in R$ satisfying the following two conditions:
- $\ell$ is a multiple of $a$ and $\ell$ is a multiple of $b$, i.e., $\ell$ is a common multiple of $a$ and $b$.
- If $x$ is a multiple of $a$ and $x$ is a multiple of $b$, then $x$ is a multiple of $\ell$.
When $R = \ZZ$, it is most common to let $LCM(a,b)$ denote the unique positive least common multiple of $a$ and $b$.
Logical Connections
This definition logically relies on the following definitions and statements: Def/Integer
The following statements and definitions logically rely on the material of this page: State/Canonical decompositions can be used to find GCD and LCM, State/Chinese remainder theorem, and State/Solutions to homogeneous linear Diophantine equations can be found with the LCM
To visualize the logical connections between this definition and other items of mathematical knowledge, you can visit any of the following clusters, and click the "Visualize" tab: Clust/Basic number theory, Clust/Factorization in rings
