Putnam 2003 B3

Show that for each positive integer n, $$n! = \prod_{i=1}^n \mathrm{lcm}\{1, 2, \dots, \lfloor n/i\rfloor\} .$$ (Here $\mathrm{lcm}$ denotes the least common multiple, and $\lfloor x \rfloor$ denotes the greatest integer $\leq x$.)