Let $(R,\cdot,+)$ be an integral domain with the multiplicative neutral element $1,$ and let $M\subseteq R$ be its finite subset. The element $a$ is called the **least common multiple** of $M,$ if and only if:

- $m\mid a\quad\forall m\in M$, i.e. all elements $m\in M$ are divisors of $a$, i.e. $a$ is a
**common multiple**of $M$, and - $m\mid a’\quad\forall m\in M\Rightarrow a\mid a’$, i.e. $a$ divides any other common multiple $a’$ of $M.$

We express these two conditions being fulfilled simultaneously for $a$ by writing $a=\operatorname{lcm}(M).$

- This definition corresponds to the special case of a least common multiple, if $R=\mathbb Z.$

| | | | | created: 2019-06-27 21:08:24 | modified: 2019-06-27 21:50:10 | by: *bookofproofs* | references: [8250]

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[8250] **Koch, H.; Pieper, H.**: “Zahlentheorie – Ausgewählte Methoden und Ergebnisse”, Studienbücherei, 1976

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