Competition problem 2025 — Number theory

, 2025

Number theory Difficulty 5.0 AIME, harder Prove it United Kingdom

This question requires answers only.

Consider the number $M = 99\ldots99$ which consists of several digit nines. A single division sign is placed between two adjacent digits of $M$ and the resulting calculation is evaluated to produce a whole number $N$ .

(a) In the case when $M$ has nine digits,

(i) How many possible values can $N$ take? [1 mark]

(ii) How many digits does the smallest possible value of $N$ have? [2 marks]

(b) In the case when $M$ has 2025 digits,

(i) How many possible values can $N$ take? [3 marks]

(ii) How many digits does the smallest possible value of $N$ have? [4 marks]

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