🧮 Chinese Remainder Theorem Calculator
Solve simultaneous congruences with step-by-step solutions
📝 Enter Congruences
❌ Error
✅ Solution Found
📐 How to Use the Chinese Remainder Theorem Calculator
- Enter pairs of remainders and moduli for your congruences
- Ensure moduli are pairwise coprime (their GCD is 1)
- Click “Calculate” to find the smallest solution satisfying all congruences
- View the step-by-step solution process
- Save or share your results
📚 What is the Chinese Remainder Theorem?
The Chinese Remainder Theorem (CRT) is a fundamental theorem in number theory that provides a method for solving simultaneous modular equations. It states that if you have a system of congruences:
- x ≡ a₁ (mod m₁)
- x ≡ a₂ (mod m₂)
- …
- x ≡ aₖ (mod mₖ)
Where the moduli m₁, m₂, …, mₖ are pairwise coprime, then there exists a unique solution modulo M = m₁ × m₂ × … × mₖ.
Applications: Cryptography, computer science, algebra, and solving complex mathematical problems.
