Problem

Source: INAMO 2023 P4 (OSN 2023)

Tags: number theory, Digits, NT construction, Indonesia, Indonesia MO



Determine whether or not there exists a natural number $N$ which satisfies the following three criteria: 1. $N$ is divisible by $2^{2023}$, but not by $2^{2024}$, 2. $N$ only has three different digits, and none of them are zero, 3. Exactly 99.9% of the digits of $N$ are odd.