Undergrad Solving congruences and diophantine equations in number theory

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SUMMARY

This discussion focuses on solving systems of congruences and Diophantine equations in number theory. Specific examples provided include the system of congruences 8x ≡ 4 (mod 20), 15x ≡ 10 (mod 35), and 9x ≡ 12 (mod 39), as well as the system x ≡ 1 (mod 15) and x ≡ 7 (mod 18). Participants seek clarification on methods to solve these equations and their applications in similar mathematical problems. The conversation emphasizes the importance of understanding modular arithmetic and the Chinese Remainder Theorem for effective problem-solving.

PREREQUISITES
  • Understanding of modular arithmetic
  • Familiarity with Diophantine equations
  • Knowledge of the Chinese Remainder Theorem
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the Chinese Remainder Theorem in detail
  • Practice solving linear Diophantine equations
  • Explore advanced techniques in modular arithmetic
  • Learn about applications of congruences in cryptography
USEFUL FOR

Mathematicians, students of number theory, and anyone interested in solving complex systems of equations and enhancing their problem-solving skills in modular arithmetic.

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Hello. I do not understand how to solve systems of three or two congruences of one unknown of first order, a congruence of one unknown of second order and a system of diophantine equations of two or three unknowns. Could someone help me by providing examples in these cases? Thank you.
 
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Why don't you post an example of an equation and what you don't understand about how to solve it
 
Solve the system 8x ≡ 4(mod20), 15x ≡ 10(mod35), 9x ≡ 12(mod39). As well, if you can tell me how to apply it to other similar situations. Find also the solutions of the system x ≡ 1(mod15), x ≡ 7(mod18). Thank you.
 
I enjoy systems of equations such as these. They remind me of a multifaceted gem.
 
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