What is Chinese Remainder Theorem Calculator?
The Chinese Remainder Theorem Calculator solves systems of simultaneous congruences efficiently. This mathematical tool finds practical applications in cryptography, computer science, and number theory. By inputting remainders and moduli values, users can instantly compute solutions to modular arithmetic problems. The CRT calculator simplifies complex calculations involved in RSA encryption, calendar computations, and error-correcting codes. Our calculator handles both pairwise coprime and non-coprime modulus cases, providing step-by-step solutions for better understanding of modular equations resolution.
CRT Calculator
Chinese Remainder Theorem Formula
The CRT formula for a system of congruences x ≡ a₁ (mod m₁), ..., x ≡ aₖ (mod mₖ) is:
X = Σ(aᵢ * Mᵢ * Nᵢ) mod M
Where M = product of moduli, Mᵢ = M/mᵢ, and Nᵢ = modular inverse of Mᵢ mod mᵢ
How to Use the Calculator
1. Enter remainder and modulus pairs in input fields
2. Click 'Add Equation' for additional congruences
3. Press 'Calculate' to get solution
4. Results show unique solution modulo product of moduli
5. Clear button resets all inputs
The calculator automatically checks pairwise coprime conditions and handles both cases, providing either unique solutions or all possible solutions when moduli aren't coprime.