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Floor function and coprime

Source: 2018 China TST 2 Day 2 Q4

January 9, 2018
number theoryfloor functionfunction

Problem Statement

Let k,Mk, M be positive integers such that k−1k-1 is not squarefree. Prove that there exist a positive real α\alpha, such that ⌊α⋅kn⌋\lfloor \alpha\cdot k^n \rfloor and MM are coprime for any positive integer nn.
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