As we started "must-to-ask" questions, see Seva's post we should ask one more natural question.

For integer $n\ge 0$, let $s(n)$ denote the sum of the digits in the decimal representation of $n$.

1) What is the smallest $M=M(N)$ such that for every pair $(a,b)$, $1\le a< b\le N$ exist $n$, $1\le n\le M$ with $s(na)\ne s(nb)$?

2) One person chooses a number $a\le N$. We can choose $k$ (one by one) and ask him about $s(ka)$. What is the least number of questions $Q=Q(N)$ we need to determine $a$?

3) The same problem as 2). But we must choose ours $k_1$,...,$k_Q$ at once.

reallybig number $k$. $\endgroup$ – fedja Apr 21 '12 at 1:06