Factor

$\begin{aligned} S_n & = 0! + 1\times 1! + 2\times 2! + 3\times 3! + \cdots + n\times n! \\ & = 0! + \sum_{k=1}^n k(k!) \\ & = 1 + \sum_{k=1}^n \big((k+1)k! - k!\big) \\ & = 1 + \sum_{k=1}^n (k+1)! - \sum_{k=1}^n k! \\ & = 1 + (n+1)! - 1! \\ & = (n+1)! \end{aligned}$

$\implies S_{2016} = 2017!$ . Yes, it is divisible by 2017.

$0!+1\times 1!+2\times 2!+...+2016\times 2016!\\ 1+1\times 1!+2\times 2!...+2016\times 2016!\\ (1+1)\times 1!+2\times 2!...+2016\times 2016!\\ 2!+2\times 2!+...+2016\times 2016!\\ (2+1)\times 2!+...+2016\times 2016!\\ 3!+3\times 3!...+2016\times 2016!=2017!$