ascheras said:ok, upon doing it (mod 7) and (mod 5), i got (3,4) (mod 7) and (1,2) (mod 7). does that sit well? or should i now apply the CRT?
A system of linear congruences is a set of equations that involve modular arithmetic, where the unknown variables are integers and the equations are congruent modulo some integer. In simpler terms, it is a collection of equations with the goal of finding integer solutions that satisfy all the equations simultaneously.
The Chinese Remainder Theorem is a theorem in number theory that provides a method for solving a system of linear congruences. It states that if the moduli of the congruences are pairwise relatively prime, then there exists a unique solution to the system. This theorem is often used to efficiently solve systems of congruences in cryptography and other areas of mathematics.
Yes, a system of linear congruences can have more than one solution. In fact, there can be infinitely many solutions depending on the moduli and equations involved. However, the Chinese Remainder Theorem guarantees that if the moduli are pairwise relatively prime, then there exists a unique solution.
The most common method for solving a system of linear congruences is through the Chinese Remainder Theorem. This involves breaking down the system into smaller, simpler congruences and using the theorem to find the unique solution. Other methods include substitution and elimination, similar to solving systems of linear equations.
Systems of linear congruences have various applications in cryptography, such as in the RSA encryption algorithm. They are also used in creating schedules and timetables, as well as in solving problems in number theory and combinatorics. Additionally, they can be applied in solving puzzles and games, such as Sudoku and KenKen.