Q: Determining Divisibility Using Euler's Criterion I am struggling with a number theory question.
The questions is whether or not $83$ divides $2^{41} - 1$.
However, I am supposed to use Euler's Criterion.
So far, I have computers the Legendre symbol for $(\frac{2}{83}) = -1$.
Then we know that $2^{\frac{83 - 1}{2}} \equiv -1 (mod 83)$
This is what I have so far, but I am not even sure if the above work is right, or how I am supposed to determine divisibility using this. 

A: Your (correct) work shows that $83$ divides $2^{41}+1$. So $83$ cannot divide $2^{41}-1$.
