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 über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927