OBM 2013 Problem 4

Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$ , $1 \le k \le n$ , the $k$ -digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$ -digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$ .

P.S.: This is basically the same problem as http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=548550.