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 modular congruences, where you need to find a number that satisfies multiple remainder conditions simultaneously. This powerful mathematical tool is essential in number theory, cryptography, and computer science applications. The calculator automates the complex computation process, making it accessible for students, researchers, and professionals working with modular arithmetic problems.
How to use
Enter your system of congruences by specifying the remainders and their corresponding moduli. Each congruence should be in the form x ≡ a (mod m). Click solve to get the unique solution modulo the product of all moduli, provided the moduli are pairwise coprime.
Frequently asked questions
What conditions must the moduli satisfy?
The moduli must be pairwise coprime (their greatest common divisor equals 1) for the Chinese Remainder Theorem to guarantee a unique solution.
How many congruences can I solve at once?
Most calculators handle 2-10 congruences simultaneously. The computation complexity increases with more equations, but the theorem works for any finite number.
What if my moduli aren't coprime?
If moduli share common factors, the system may have no solution or multiple solutions. The classical Chinese Remainder Theorem doesn't apply directly.