Prove $\, a_1 \Bbb Z \cap \dotsb \cap a_r \mathbb{Z} = {\rm lcm}(a_1, \ldots, a_r) \Bbb Z\ $ [lcm = ideal intersection]

$$c \in a_1\Bbb Z\cap\cdots\cap a_r\Bbb Z\iff a_1,\ldots,a_r\mid c\!\!\color{#c00}{\overset{\rm U\!}\iff} \overbrace{{\rm lcm}(a_1,\ldots,a_r)}^{\large \ell}\mid c\iff c\in\ell\,\Bbb Z\quad $$

where $\,\rm\color{#c00}U = $ Universal Property of LCM (which is the definition of lcm in general domains).

Remark $ $ Above is simply the PID case of the universal property of (ideal) intersection, i.e.

$\qquad\ \ A_1 \cap \cdots \cap A_n \supseteq C\iff A_1\supseteq C,\ldots, A_n\supseteq C$.

As such, one often defines $\,{\rm lcm}(A_1,\ldots,A_n) := A_1 \cap \cdots \cap A_n,\ $ for ideals $\,A_i,$

and, $ $ dually, $ $ one $ $ defines $ \ \gcd(A_1,\ldots,A_n) := A_1 + \cdots + A_n$

since in a PID $\,A\supseteq C\iff (a)\supseteq (c)\iff a\mid c,\,$ i.e. contains = divides.

Generally we don't have $\,{\rm lcm}(A,B)\gcd(A,B) = AB\,$ for $\,A,B\neq 0,\,$ but this does hold true in a broad class of domain known as Prufer domains - which are non-Noetherian generalizations of Dedekind domains. Their ubiquity stems from a remarkable confluence of interesting characterizations. For example, they are those domains satisfy either the Chinese Remainder Theorem for ideals, or Gauss's Lemma for polynomial content ideals, or for ideals: $\, A\cap (B + C) = A\cap B + A\cap C\,\, $ or "contains = divides" $\, A\supset B\ \Rightarrow\ A\mid B\,$ for fin. gen. $A\,$ etc. It has been estimated that there are close to $100$ such characterizations known. Here is around $30$ interesting ones.

The ring $\Bbb Z$ of integers is a principal ideal ring. Therefore, we may choose unique $b\in\Bbb N$ such that $$ b\Bbb Z = a_1\Bbb Z\cap a_2\Bbb Z\cap ... \cap a_r\Bbb Z$$ implying $b$ must be a common multiple of $a_1,a_2,...,a_r$ whose least common multiple we denote $b'$ so that $b'\;|\;b$ and $$b'\Bbb Z \subset (a_1\Bbb Z\cap a_2\Bbb Z\cap ... \cap a_r\Bbb Z) = b\Bbb Z$$ implying $b\;|\;b'$ and therefore $b=b'$.