Find the necessary and sufficient conditions for all $ 41 \mid \underbrace{11\ldots 1}_{n}$, $n\in N$. Find the necessary and sufficient conditions for all $ 41 \mid \underbrace{11\ldots 1}_{n}$, $n\in N$. And, if $\underbrace{11\ldots 1}_{n}\equiv 41\times p$,
then $p$ is a prime number. 
Find all of the possible values of $n$ to satisfy the condition.

The first sentence is asking that $41|\frac {10^n-1}9$.  This is just the length of the repeat of $\frac 1{41}$.  The second statement forces $n$ to be the minimum value.  Without it, any multiple of $n$ would work, but $p$ would not be prime.
