Let $ k,M$ be positive integers such that $ k−1$ is not squarefree. Prove that there exist a positive real number $ α$ , such that $ ⌊α⋅k^n⌋$ and M are coprime for any positive integer $ n$ .

Since $ k−1$ is not squarefree, then there exists a prime number $ p$ such that $ p^2|k−1$ . Choose $ α=N+\frac{1}{p}$ , with $ N$ is a positive integer such that $ p⋅N+1$ is divisible by all prime factors of $ M$ (except $ p$ if $ p|M$ ), and $ N$ is not divisible by $ p$ . (we can choose $ N$ by using Chinese Remainder Theorem). Then for every positive integer $ n$ , $ $ ⌊α⋅k^n⌋=N⋅k^n+⌊\frac{k^n}{p}⌋=N⋅k^n+\frac{k^n−1}{p}=\frac{k^n⋅(p⋅N+1)−1}{p} $ $ Since $ p⋅N+1$ is divisible by all prime divisors of $ M$ , and $ ⌊α⋅k^n⌋$ is not divisible by $ p$ , because $ N⋅k^n$ is not divisible by $ p$ (we consider this in case $ p|M$ ), therefore, $ ⌊α⋅k^n⌋$ and $ M$ are coprime.

However, if $ α$ must be irrational, then I have a feeling that there are no such $ α$ that suit the problem’s condition.

So my question is:

Let $ p$ be a prime integer, $ k$ be a positive integer and $ α$ be a positive irrational number. Is it true that there always exists a positive integer $ n$ such that $ p | ⌊k^n⋅α⌋$ ?

Any answers or comments will be appreciated.

(Please let me know if this question should be closed, off-topic or unclear. I may not visit this page frequently, so I may not be able to know what is going on. Sorry for this inconvenience)