Unless I am mistaken, for any $a, n \in \mathbb N$ the following are equivalent:
$\text{LCM}(a, n)/a$
The minimal $\ell \in \mathbb N$ such that $\underbrace{a + a + \cdots + a}_{\ell \text{ times}}$ is divisible by $n$
The minimal $\ell \in \mathbb N$ such that $a \cdot \ell \equiv 0\ (\text{mod }n)$
Does this value $\ell = \ell(a, n)$ have a common name? I have been unable to find one searching online. It's similar to multiplicative order but not quite the same.
(Tagged group-theory due to its similarity with multiplicative order)
The beauty of abstracting concepts from elementary mathematics is that many seemingly different notions can turn out to fall under one unifying term. Your question falls into that phenomenon.
In elementary number theory, the order of $a$ modulo a prime $p$ is the smallest natural number $m$ such that $a^m \equiv 1 \pmod{p}$. This translates to $\operatorname{ord}(a) = m$ in $(\mathbb{F}_p^{\times}, \cdot)$ which motivates the order of a group element $g$ in a group $G$: It 's the smallest natural number $m$ such that $g^m = e$.
Specializing again, the smallest natural number $m$ such that $am \equiv 0 \pmod n$ is the order of $a$ in the group $(\mathbb{Z}/n \mathbb{Z}, +)$. Sometimes, one says that this is the additive order of $a$ if the group operation is not clear.