USAMO 1972 Problem 1

Let $(\mathrm{a}, \mathrm{b}, \ldots, \mathrm{k})$ denote the greatest common divisor of the integers $\mathrm{a}, \mathrm{b}, \ldots \mathrm{k}$ and $[\mathrm{a}, \mathrm{b}, \ldots, \mathrm{k}]$ denote their least common multiple. Show that for any positive integers $\mathrm{a}, \mathrm{b}, \mathrm{c}$ we have $(\mathrm{a}, \mathrm{b}, \mathrm{c})^{2}[\mathrm{a}, \mathrm{b}][\mathrm{b}, \mathrm{c}][\mathrm{c}, \mathrm{a}]=[\mathrm{a}, \mathrm{b}$, $c]^{2}(a, b)(b, c)(c, a)$.