On the sum of digits of the factorial

Let b > 1 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, where C_b is a positive constant depending only on b. This improves of 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,...,n.