From my previous question: Is it true that there always exists a positive integer $n$ such that $p | ⌊k^n⋅α⌋$ , I came up with a similar question:

Given a positive integer $k$ such that $k>2$, $k-1$ is not squarefree. What are the conditions of the positive integer $M$ so that there exists a positive irrational number $\alpha$ such that for every positive integer $n$, $\lfloor k^n \alpha \rfloor$ and $M$ are always coprime?

If $gcd(M,k)=1$, then let $s=ord_M(k)$. Let $a_1,a_2,...$ be positive integers less than $k$ such that $gcd(a_1+a_2+...+a_i,M)=1$ for every $i$. Choose $$\alpha=\frac{a_1}{k^s}+\frac{a_2}{k^{3s}}+...+\frac{a_j}{k^{\frac{sj(j+1)}{2}}}+...=\overline{0.00...0a_10...0a_20...}_k$$ in base k.

Then for every $n$, $\lfloor k^n \alpha \rfloor \equiv k^l(a_1+a_2+...+a_i) \mod(M)$ for some positive integer $l$ and $i$, thus satisfy the condition.

If $M$ is a prime number such that $M|k$, then let $c$ be a positive integer different from $1$, is not divided by $M$ and less than $ k$. Choose $$\alpha=\overline{0.1c11cc111ccc...}_k$$ in base k.

Then for every $n$, $\lfloor k^n \alpha \rfloor \equiv 1$ or $c$$\mod(M)$, therefore $\lfloor k^n \alpha \rfloor $ and $M$ are coprime.

However, if $M$ and $k$ are not coprime and $M$ is not a prime, then are there any conditions so that for every positive integer $n$, $\lfloor k^n \alpha \rfloor$ and $M$ are always coprime ?

Moreover, are my solutions above correct? More specifically, if, for example, $\overline{0.abcde...}_{10}$ (decimal form) is irrational, then is $\overline{0.abcde...}_k$ (in base $k$) coprime ?

(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. Any answers or comments will be appreciated)