To complete my answer to this question, one needs to prove the following conjecture that seems to be true:
- For every two natural numbers $a \lt b$, $a,b\not\equiv 0\mod 10$, there exists $n\lt b$ such that the sums of decimal digits of $na, nb$ are different.
Note that we cannot claim $n\le a$ because for $a=10^k+1, b=10^{k+1}+1$, the smallest such $n$ is $10^k+9$ (for every $k \ge 1$).
Update It looks like if $n=n(a,b)$ exceeds $b/10$, then $a$ and $b$ (if big enough) must have a very specific form: all but two digits are 0's or all but 2 digits are 9's. If that is true, my conjecture will be true as well. In fact even if $n(a,b)\ge \sqrt{b}$, then $a$ and $b$ look very similar.
