RIOPLATENSE 2016 Problem 6

When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits $123456789101112 ....$ Denote $A_k$ the number formed by the first $k$ digits of this sequence . Prove that for all positive integer $n$ there is a positive integer $m$ which simultaneously verifies the following three conditions: (i) $n$ divides $A_m$ , (ii) $n$ divides $m$ , (iii) $n$ divides the sum of the digits of $A_m$ .