  Let $\gcd(d_{1},\ldots,d_{k})$ be the greatest common divisor of the positive
integers $d_{1},\ldots,d_{k}$, for any integer $k\geq 2$, and let $\tau$ and
$\mu$ denote the divisor function and the M\"{o}bius function, respectively.
For an arbitrary arithmetic function $g$ and for any real number $x>5$ and any
integer $k\geq 3$, we define the sum $$ S_{g,k}(x) :=\sum_{n\leq
x}\sum_{d_{1}\cdots d_{k}=n} g(\gcd(d_{1},\ldots,d_{k})) $$ In this paper, we
give asymptotic formulas for $S_{\tau,k}(x)$ and $S_{\mu,k}(x)$ for $k\geq 3$.
