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How do I do this question using the pigeonhole principle? Of course, I could just list down values of $k$ such that $2^k-1$ is divisible by 11. For example $k = 10$ would nicely solve the question but it requires listing and when the divisor gets bigger (For example show that there exists a $k$ where $2^k-1$ is divisible by 21, it's harder and much more complicated to solve it.
Are there any hints how I can use pigeonhole principle to solve this question?