On the sum of digits of the factorial

Let $b \geq 2$ be an integer and denote by $s_b(m)$ the sum of the digits of the positive integer $m$ when is written in base $b$. We prove that $s_b(n!) > C_b \log n \log \log \log n$ for each integer $n > e^e$, where $C_b$ is a positive constant depending only on $b$. This improves by a factor $\log \log \log n$ a previous lower bound for $s_b(n!)$ given by Luca. We prove also the same inequality but with $n!$ replaced by the least common multiple of $1,2,\ldots,n$.