Competition problem — Number theory

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Number theory Difficulty 5.3 AIME, harder Prove it Ukraine

Given natural number $N$ we write two different nonzero digits from the right. It turns out, that new number is divisible by $N$ . What is maximal value of $N$ can be?

Solution

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.

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