BALTIC-WAY 2007 Problem 19

Let $r$ and $k$ be positive integers such that all prime divisors of $r$ are greater than $50$ .

A positive integer, whose decimal representation (without leading zeroes) has at least $k$ digits, will be called nice if every sequence of $k$ consecutive digits of this decimal representation forms a number (possibly with leading zeroes) which is a multiple of $r$ .

Prove that if there exist infinitely many nice numbers, then the number $10^k-1$ is nice as well.