fractions – Numbers and the period of their inverse

I am currently investigating the correlation between a natural number and the length of its inverse’s period. For example:

$$
frac{1}{3760} = 0.00026595744680851063829787234042553191489361702127 (period 46) \
frac{1}{1122} = 0.00089126559714795 (period 16)
$$

And so on.
There is a simple explanation for why does this happen mixing powers and modular arithmetic but it’s just an iterative process, which is troublesome for really big numbers.

Which conjectures, theories, hypothesis exists regarding the computation of the periodic length of any natural number's inverse?