Question 169
Prove that for any positive integer $n$, we have
$$
\prod_{k=1}^{n} \operatorname{lcm}\left(1,2, \ldots,\left\lfloor\frac{n}{k}\right\rfloor\right)=n !
$$
$$
\prod_{k=1}^{n} \operatorname{lcm}\left(1,2, \ldots,\left\lfloor\frac{n}{k}\right\rfloor\right)=n !
$$