# The digit sum: $s(na)=s(nb)$
For integer $n\ge0$, let $s(n)$ denote the sum of the digits in the decimal representation of $n$.
Is it true that for any integer $a,b>0$, the ratio of which is not a power of $10$, the set of all those $n\ge 0$ with $s(an)=s(bn)$ has zero density?