It is a known result from set theory that in contrast to the ordered set of integers $(\mathbb Z,\le)$, the ordered set of natural numbers $(\mathbb N,\le)$ is well-ordered. This property of natural numbers ensures the existence of a smallest element in any non-empty subset of natural numbers. The existence is very useful in the study of divisibility, therefore, in the following, we will restrict our concept of divisibility to natural numbers only.

Proposition: Least Common Multiple

Let \(a, b\in\mathbb Z\) be integers and let \(M_{a,b}\) be the set of all common multiples of \(a\) and \(b\):

\[M_{a,b}:=\left\{m\in\mathbb Z : a\mid m\wedge b\mid m\right\}.\]

\(M_{a,b}\) has a minimum $\min(M_{a,b})$, called the least common multiple of \(a\) and \(b\), denoted by

\[\operatorname{lcm}(a,b):=\min(M_{a,b}).\]

Moreover, $\operatorname{lcm}(a,b)\mid m\in M_{a,b}$, i.e. \(\operatorname{lcm}(a,b)\) is a divisor of any other common multiple of \(a\) and \(b\).

Proofs: 1

Definitions: 1
Proofs: 2 3 4 5 6 7
Propositions: 8 9 10 11 12 13


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References

Bibliography

  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927