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).
explanation: The text extract contains a math question and a partial attempt at solving it. The question asks to find an \( x \) such that \( 5003 \) divides \( x^2 + 10 \), or to prove that no such \( x \) exists. The user provides some initial steps and mentions the use of Euler's criterion and the Euler-Gauss quadratic reciprocity law. However, the extract does not contain a complete answer to the question.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: 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).
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract contains a math question and a partial attempt at solving it. The question asks to find an \( x \) such that \( 5003 \) divides \( x^2 + 10 \), or to prove that no such \( x \) exists. The user provides some initial steps and mentions the use of Euler's criterion and the Euler-Gauss quadratic reciprocity law. However, the extract does not contain a complete answer to the question.
  
  Label: question-and-partial-answer