  Let $L_n(k)$ denote the least common multiple of $k$ independent random
integers uniformly chosen in $\{1,2,\ldots ,n\}$. In this note, using a purely
probabilistic approach, we derive a criterion for the convergence in
distribution as $n\to\infty$ of $\frac{f(L_n(k))}{n^{rk}}$ for a wide class of
multiplicative arithmetic functions~$f$ with polynomial growth $r>-1$.
Furthermore, we identify the limit as an infinite product of independent random
variables indexed by prime numbers. Along the way, we compute the generating
function of a trimmed sum of independent geometric laws, occurring in the above
infinite product. This generating function is rational; we relate it to the
generating function of a certain max-type Diophantine equation, of which we
solve a generalized version. Our results extend theorems by Erd\H{o}s and
Wintner (1939), Fern\'{a}ndez and Fern\'{a}ndez (2013) and Hilberdink and
T\'{o}th (2016).
