Chinese Remainder Theorem Solver
Solve systems of modular congruences using the Chinese Remainder Theorem
About this calculator
The Chinese Remainder Theorem Solver helps you find solutions to systems of simultaneous modular congruences. This powerful mathematical tool is essential for number theory, cryptography, and computer science applications. The calculator automatically determines the unique solution modulo the product of all moduli, provided they are pairwise coprime. It's particularly useful for RSA encryption, coding theory, and solving complex mathematical problems involving remainders.
How to use
Enter your system of congruences by inputting the remainder values and their corresponding moduli. Ensure all moduli are pairwise coprime (their greatest common divisor is 1). Click solve to get the unique solution that satisfies all congruences simultaneously, displayed as the smallest positive integer solution.
Frequently asked questions
What happens if the moduli are not coprime?
The Chinese Remainder Theorem only works when all moduli are pairwise coprime. If they share common factors, no unique solution exists.
How many congruences can I solve at once?
Most calculators handle 2-10 congruences simultaneously. The computation becomes more complex with larger systems but remains mathematically valid.
What are practical applications of this theorem?
Common uses include RSA cryptography, computer memory addressing, calendar calculations, and solving polynomial congruences in number theory research.