On the sum of digits of the factorial

Let b≥2 be an integer and denote by sb(m) the sum of the digits of the positive integer m when is written in base b. We prove that sb(n!)>Cblog⁡nlog⁡log⁡log⁡n for each integer n>ee, where Cb is a positive constant depending only on b. This improves by a factor log⁡log⁡log⁡n a previous lower bound for sb(n!) given by Luca. We prove also the same inequality but with n! replaced by the least common multiple of 1,2,…,n.