UkraineMO

Number theory Difficulty 5.5 AIME, harder Prove it Ukraine

Find all positive integers $n$ , such that $11^n - 1$ is divisible by $10^n - 1$ .

Solution

Since $10^n - 1 = 9 \cdot (10^{n-1} + \dots + 10 + 1)$ we obtain that $11^n - 1$ is divisible by $9$ . Considering the residues modulo $9$ : $11^n - 1 \equiv 2^n - 1 \pmod{9}$ we get $n = 6k$ . But then $10^{6k} - 1$ is divisible by $10^6 - 1$ . Hence it is divisible by $10^3 + 1$ and $10 + 1 = 11$ which is not possible.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.