Let $(ξ_1, η_1)$, $(ξ_2, η_2),\ldots$ be independent identically distributed $\mathbb{N}^2$-valued random vectors with arbitrarily dependent components. The sequence $(Θ_k)_{k\in\mathbb{N}}$ defined by $Θ_k=Π_{k-1}\cdotη_k$, where $Π_0=1$ and $Π_k=ξ_1\cdot\ldots\cdot ξ_{k}$ for $k\in\mathbb{N}$, is called a multiplicative perturbed random walk. We study arithmetic properties of the random sets $\{Π_1,Π_2,\ldots, Π_k\}\subset \mathbb{N}$ and $\{Θ_1,Θ_2,\ldots, Θ_k\}\subset \mathbb{N}$, $k\in\mathbb{N}$. In particular, we derive distributional limit theorems for their prime counts and for the least common multiple.