MathDB

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.

Problem Statement