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Let $B_n(m)$ be a set picked uniformly at random among all $m$-elementssubsets of $\{1,2,\ldots,n\}$. We provide a pathwise construction of thecollection $(B_n(m))_{1\leq m\leq n}$ and prove that the logarithm of the leastcommon multiple of the integers in $(B_n(\lfloor mt\rfloor))_{t\geq 0}$,properly centered and normalized, converges to a Brownian motion when both$m,n$ tend to infinity. Our approach consists of two steps. First, we show thatthe aforementioned result is a consequence of a multidimensional central limittheorem for the logarithm of the least common multiple of $m$ independentrandom variables having uniform distribution on $\{1,2,\ldots,n\}$. Second, weoffer a novel approximation of the least common multiple of a random sample bythe product of the elements of the sample with neglected multiplicities intheir prime decompositions.

A Brownian weak limit for the least common multiple of a random m-tuple of integers