On the least common multiple of $q$-binomial coefficients

In this paper, we prove the following identity $$\text{lcm}({\left[\genfrac{}{}{0px}{}{n}{0}\right]}_q,{\left[\genfrac{}{}{0px}{}{n}{1}\right]}_q,\mathrm{.}\mathrm{.}\mathrm{.},{\left[\genfrac{}{}{0px}{}{n}{n}\right]}_q)=\frac{\text{lcm}([1{]}_q,[2{]}_q,\mathrm{.}\mathrm{.}\mathrm{.},[n+1{]}_q)}{[n+1{]}_q},$$ where ${\left[\genfrac{}{}{0px}{}{n}{k}\right]}_q$ denotes the $q$-binomial coefficient and $[n{]}_q=\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)].