text: Hi,
  
  I am not too good at this stuff, but getting better haha..
  
  Anyway, find an x which satisfies the statement 5003 divides x^2 + 10, or else prove that no such x exists.
  
  I started by switching it around to say x^2 = -10 mod 5003.
  I kind of suspect there is no solution, and there is a theorem (euler's criterion) which states that -10 is a quadratic residue (i.e. there is a solution) if and only if (-10)^(5002/2) = (-10)^2501 = 1 mod 5003.
  
  (-10)^2501 is a huge number tho, and i'm not sure how to deal with it. Any hints would be great.
  
  2. You need to use the Euler-Gauss quadratic reciprocity law, along with the fact that $\displaystyle \left(\frac{-10}{5003}\right)=\left(\frac{-1}{5003}\right)\left(\frac{2}{5003}\right)\left(\f rac{5}{5003}\right)$. (Assuming you are familiar with this notation for the Legendre symbol).