Prove: for all $n$ there's a $m$ such that the sum of digits in $mn$ is equal to $n$.

Question

In the following $n,m$ are natural numbers.

I need to prove that for all $n$ there's a $m$ such that the sum of digits in $mn$ is equal to $n$.

Any ideas?

Thanks.

Answer 1

A proof is in this paper of H. Fredricksen, E. J. Ionascu, F. Luca, and P. Stanica. See also OEIS sequence A131382.

Answer 2

There is an infinite amount of powers of $10$, however only a finite amount of congruence classes mod $m$, hence there must be at least one such congruence class containing an infinite amount of powers of $10$. Take $n$ distinct powers of ten in this congruence class. The sum of these numbers is a multiple of $n$ and the digits in the base $10$ representation of the sum are all zeroes except for $n$ ones, thus the sum of the digits is $n$.