Length of cycle What is the length of the cycle of 1/71?
Im basically stuck at this part of a long question. I dont know what I am suppose to find and my text book have no examples of it.
 A: It might help you to know that this is one of the many wonderful things that Wolfram Alpha can do for you.  In this case, the answer is $35$, but why is that the case?
As a hint: notice that $71$ is prime, and that the period divides $70=71-1$.  This is not a coincidence.
A: Hint: As has been pointed out, appropriate software handles the problem quickly. Finding the answer by hand can be done, though it is more painful.
We want to compute the order of $10$ modulo $71$. The candidates are the divisors of $70$. 
To compute less, if you are accustomed to Legendre symbol calculations, you will quickly find that $(10/71)=1$. So $10$ is a quadratic residue of $71$, and therefore cannot have order $70$, and indeed (since $70$ is not divisible by $4$), the order of $10$ cannot be even. That leaves the possibilities $5$, $7$, and $35$. We can use a simple calculator to show that $71$ does not divide $10^5-1$ or $10^7-1$. So the order must be $35$.
A: Wolfram says period is 35.  Is this what you meant to ask?
$$\frac{1}{71}=0.0\overline{14084507042253521126760563380281690}$$
