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 \(% \mathbb{U}_{N}\) of positive integers less than \(N\) and relatively primes to \(% N,\) 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 \(% d\geq2\) and such that \(|b|_{N}=kd\) and \(\frac{x}{N}=0.% bar{a_{1}a_{2}...a_{|b|_{N}}}\) 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 \(% S_{d}(x)=\sum\limits_{j=1}^{d}A_{j}\). 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 \(d\).