The result that is given in Wikipedia without proof is an immediate corollary of the following:
Theorem: Let $D$ be a domain and $a,b\in D$. TFAE $\colon$
i) $\text{lcm}(a,b)$ exists.
ii) For all $r\in D\setminus\{0\}$, $\gcd(ra,rb)$ exists.
Proof: i)$\implies$ ii) Let's call $\text{lcm}(a,b)=m$. We have $a\mid ab$ and $b\mid ab$, then $m\mid ab$. We claim that $\gcd(a,b)=ab/m$. Indeed, $$a=\frac{ab}{m}\frac{m}{b}\implies \frac{ab}{m}\mid a,$$ $$b=\frac{ab}{m}\frac{m}{a}\implies \frac{ab}{m}\mid b.$$
Thus, $ab/m$ is a common divisor of $a$ and $b$.
Let $e\in D$ such that $e\mid a$ and $e\mid b$. Then, $e \mid ab$, so that $ab/e$ is an integer. Moreover, $a \mid ab/e$ (since $e \mid b$) and $b \mid ab/e$ (likewise), whence $m \mid ab/e$. That is, $em\mid ab$. Hence $e\mid ab/m$, and this shows that $\gcd(a,b)=ab/m$.
Set $\gcd(a,b)=ab/m=d$. Given $r\neq 0$, we claim that $\gcd(ra,rb)=rd$. Indeed, $d\mid a$ and $d\mid b$ implies that $rd\mid ra$ and $rd\mid rb$. Let $s\in D$ such that $s\mid ra$ and $s\mid rb$. Then, $rab/s$ is an integer and satisfies $a \mid rab/s$ (since $s \mid rb$) and $b \mid rab/s$ (likewise), so that $m \mid rab/s$. Hence, $sm\mid rab$. Therefore $s\mid rab/m=rd$. This proves that $\gcd(ra,rb)=rd$.
ii)$\implies$ i) We claim that $$\text{lcm}(a,b)=\frac{ab}{\gcd(a,b)}.$$
Indeed, set $d=\gcd(a,b)$, then $$\frac{ab}{d}=a\frac{b}{d},$$ $$\frac{ab}{d}=b\frac{a}{d}.$$
So $a\mid ab/d$ and $b\mid ab/d$. Let $n\in D$ such that $a\mid n$ and $b\mid n$, thus $ab\mid nb$ and $ab\mid na$. It follows that $ab\mid \gcd(na,nb)=n\gcd(a,b)$, i.e., $ab\mid nd$ and then $\frac{ab}{d}\mid n$. Hence, $\text{lcm}(a,b)=ab/d$.
The above theorem is used in this paper (Dinesh Khurana, On GCD and LCM in domains -- A conjecture of Gauss, Resonance, 8(6), 72–79) by D. Khurana where he proves that for every $n\ge 3$ non-square $\Bbb{Z}[\sqrt{-n}]$ is not a GCD-domain, and hence not a UFD. More exactly, he shows that if $n+1=pk$ for some prime $p$ and $k\ge 2$, then $\text{lcm}(p,1+\sqrt{-n})$ doesn't exist; and if $n+1$ is prime then $\text{lcm}(2,2+\sqrt{-n})$ doesn't exist. So we have an alternative proof to the fact that $\Bbb{Z}[\sqrt{-n}]$ is not a UFD for $n\ge 3$.
