Proof

Let $a,b,c\in\mathbb Z$ be integers.
The commutativity of $\operatorname{lcm}$ follows immediatly from the definition of the least common multiple because of the associativity of conjunction; i.e. since $a\mid m\wedge b\mid m$ if and only if $b\mid m\wedge a\mid m,$ the sets $M_{a,b}$ and $M_{b,a}$ are equal and have the same minimum.
The associativity of $\operatorname{lcm}$ can be understood as follows:
- If $m$ is a multiple of both, $\operatorname{lcm}(a,b)\mid m$ and $c\mid m,$ then $\operatorname{lcm}(\operatorname{lcm}(a,b),c))\mid m.$ Moreover, $a\mid m$ and $b\mid m.$
- Therefore, $m$ is also a multiple of both, $a\mid m$ and $\operatorname{lcm}(b,c)\mid m.$ Thus $m$ is a multiple of $\operatorname{lcm}(a,\operatorname{lcm}(b,c)).$
- Since this reasoning holds for any common multiple of $a\mid m,$ $b\mid m,$ and $c\mid m,$ it follows $\operatorname{lcm}(\operatorname{lcm}(a,b),c))=\operatorname{lcm}(a,\operatorname{lcm}(b,c)).$
Since $\operatorname{lcm}$ is both, commutative and associative, the notation $\operatorname{lcm}(a,b,c)$ makes sense.
By induction over $k,$ the formula $\operatorname{lcm}(n_1,\ldots,n_k)=\operatorname{lcm}(\operatorname{lcm}(n_1,\ldots,n_{k-1}),n_k)$ follows.
∎

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Scheid Harald: "Zahlentheorie", Spektrum Akademischer Verlag, 2003, 3rd Edition
Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927