I wanted to know if there is a way to easily determine the lowest value Natural number $n$ to where $1/n$ decimal actual period is purely repeating to $5$ digits in base $5$ and purely repeating to $4$ digits in base $6$?
What I learned so far
I had seen where $n$’s decimal representation independent of the base if the fraction is in lowest terms (is why I choose $1/n$) the period is limited to some $n-1$ or an integer $n-1$ is divisible by. It does not give the way to actually determine the period and it states the period may well be different for other bases.
For example $1/33$ could have periods of ($1,2,4,8,16,$ or $32$).
So my guessing point is lowest $n$ would need to be some multiple of $20$ then adding one also not having $2,3$ or $5$ as a prime factor to insure purely repeating for all chosen bases.
However I also saw for the denominator of $33$ that a period length of $10$ was given even though $33-1$ has no factors of $5$ in it so am left wondering if that theorem is actually true.