51st Ukrainian National Mathematical Olympiad, 3rd Round

Assume that two digits form the number $\overline{ab}$. Then we have $\overline{Nab}:N$, or $100N+\overline{ab}:N$, which implies $\overline{ab}:N$. Since $\overline{ab}$ is form by two distinct digits then $a\le 98$, hence $N\le 98$. The number $N=98$ satisfies all the requirements.