Find the necessary and sufficient conditions for all $ 41 \mid \underbrace{11\ldots 1}_{n}$, $n\in N$.

Question

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.

What are your thoughts? – hmakholm left over Monica Mar 22 '13 at 13:36 — hmakholm left over Monica, Mar 22 '13 at 13:36

score 0 · Answer 1 · answered Mar 22 '13 at 13:37

0

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.

answered Mar 22 '13 at 13:37

Ross Millikan

383,099

I could try that: for the case, $11111=41\times 271$. – Thomas You Mar 22 '13 at 13:43

Find the necessary and sufficient conditions for all $ 41 \mid \underbrace{11\ldots 1}_{n}$, $n\in N$.

1 Answers1