  We use elementary arguments to prove results on the order of magnitude of
certain sums concerning the gcd's and lcm's of $k$ positive integers, where
$k\ge 2$ is fixed. We refine and generalize an asymptotic formula of
Bordell\`{e}s (2007), and extend certain related results of Hilberdink and
T\'oth (2016). We also formulate some conjectures and open problems.
