For m > 2 let $\nu_\lambda^{m}$ be the distribution of the random sum $\sum_{n=0}^{\infty } \theta_n\lambda^n$, where $\theta_n$ are i.i.d. and for every n the random variable $\theta_n$ takes values in the set {0,1,...,m-1} with equal probabilities. As a generalization of Solomyak's Theorem (in which m=2) we prove that for Lebesgue a.e. $\lambda$ in (1/m,1) the measure $\nu_\lambda^{m}$ is absolute continuous w.r.t. the Lebesgue measure. (For smaller $\lambda$, the measure $\nu_\lambda^{m}$ is supported on a Cantor set, so if $\lambda < 1/m $ then $\nu_\lambda^{m}$ is singular.) This is joint work with Hajnalka T\'{o}th.