Structure of associated sets to Midy's Property

Let $b$ be a positive integer greater than 1, $N$ a positive integer relatively prime to $b$, $|b|_{N}$ the order of $b$ in the multiplicative group $$ of positive integers less than $N$ and relatively primes to $$ and $x\in\mathbb{U}_{N}$. It is well known that when we write the fraction $\frac{x}{N}$ in base $b$, it is periodic. Let $d,\,k$ be positive integers with $$ and such that $|b|_{N}=kd$ and $\frac{x}{N}=0.$ with the bar indicating the period and $a_{i}$ are digits in base $b$. We separate the period ${a_{1}a_{2}... a_{|b|_{N}}}$ in $d$ blocks of length $k$ and let $A_{j}=[a_{(j-1)k+1}a_{(j-1)k+2}...a_{jk}]_{b}$ be the number represented in base $b$ by the $j-th$ block and $$. If for all $x\in\mathbb{U}_{N}$, the sum $S_{d}(x)$ is a multiple of $b^{k}-1$ we say that $N$ has the Midy's property for $b$ and $d$. In this work we present some interesting properties of the set of positive integers $d$ such that $N$ has the Midy's property for $b$ and $d$.

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