TITLE: Calculate the 146th digit after the decimal point of $ \frac{1}{293} $
QUESTION [18 upvotes]: The question is: Calculate the 146th digit after the decimal point of $\frac{1}{293}$
1 / 293 = 0,00341296928.., so e.g., the fifth digit is a 1.
We know that 293 is a prime, probably this would help us.
I think an equation involving modulos has to be solved, but I am not sure how to tackle this.
Any help is appreciated! Could perhaps someone give a general method to solve these kind of problems?
EDIT: You are supposed to solve this without using a computer.

REPLY [4 votes]: The following is a small variant of the methods of Sasha and Lopsy.  We show that $10$ is a quadratic residue of $293$. This enables us to conclude that $10^{146} \not\equiv -1 \pmod{293}$.
It is just a Legendre symbol calculation. For ease of typing we denote the Legendre symbol by $(a/p)$ instead of $\left(\frac{a}{p}\right)$.
Note that $(10/293)=(2/293)(5/293)$.  Since $293$ is of the form $8k+5$, we have $(2/293)=-1$. 
To calculate $(5/293)$ we use  Quadratic Reciprocity. Since one, and indeed both, of $5$ and $293$ are of the shape $4k+1$, 
$$(5/293)=(293/5)=(3/5).$$ 
One could continue with Quadratic Reciprocity, but by inspection $(3/5)=-1$. 
Thus $(10/293)=(-1)(-1)=1$, and we conclude that $10$ is indeed a quadratic residue of $293$.