KöMaL 2012 B4493

Let ($n$,$k$) denote the greatest common divisor of the positive integers $n$ and $k$, and let [$n$,$k$] denote their least common multiple. Show that, for all positive integers $a$, $b$, $c$, the greatest common divisor of the numbers [$a$,$b$], [$b$,$c$], [$c$,$a$] equals the least common multiple of the numbers ($a$,$b$), ($b$,$c$), ($c$,$a$).