Chinese Remainder Theorem Calculator
Solve systems of modular congruences using the Chinese Remainder Theorem
About this calculator
The Chinese Remainder Theorem Calculator solves systems of modular congruences where you need to find a number that satisfies multiple remainder conditions simultaneously. This powerful mathematical tool is essential for number theory, cryptography, and computer science applications. It's particularly useful for solving problems involving periodic patterns, RSA encryption algorithms, and computational mathematics where you need to work with large integers and their remainders across different modular systems.
How to use
Enter your system of congruences by specifying the remainder and modulus for each equation (e.g., x ≡ a₁ (mod m₁), x ≡ a₂ (mod m₂)). Ensure all moduli are pairwise coprime for the theorem to apply. Click calculate to find the unique solution modulo the product of all moduli.
Frequently asked questions
What does pairwise coprime mean?
Pairwise coprime means every pair of moduli has no common factors other than 1, which is required for the Chinese Remainder Theorem to work.
Can I solve systems with non-coprime moduli?
No, the Chinese Remainder Theorem only works when all moduli are pairwise coprime. Non-coprime systems may have no solution or multiple solutions.
What's the maximum number of congruences I can solve?
Most calculators handle 2-10 congruences efficiently. The computation complexity increases with more equations, but there's no theoretical limit for the theorem itself.