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 UN of positive integers less than N and relatively primes to N, and x E UN. It is well known that when we write the fraction x/ N in base b, it is periodic. Let d, k be positive integers with d > 2 and such that |b|N = dk and x /N = 0.a1a2 a|b|N with the bar indicating the period and ai are digits in base b. We separate the period a1a2 a|b|N in d blocks of length k and let Aj = [a(j 1)k+1a(j 1)k+2 ajk]b be the number represented in base b by the j - th block and Sd(x) = d Zigma j=1 Aj . If for all x E UN, the sum Sd(x) is a multiple of bk - 1 we say that N has 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 Midy's property to for b and d.