UNLOCKING NUMBER SECRETS Chinese Remainder Theorem Solver
Enter your congruences below and discover the hidden solution with ease!
Enter Congruences
Input each congruence in the form x ≡ a (mod m). Add more congruences as needed.
x ≡ (mod )
Solution
The solution is: \(x \equiv \\pmod{N}\)
Where N is the product of all moduli.
System of Congruences:
- Congruence : \( {x \equiv \pmod{}} \)
Understanding the Chinese Remainder Theorem
Imagine you have a system of equations that describe remainders when a number is divided by different moduli. The Chinese Remainder Theorem (CRT) steps in to solve these! It guarantees a unique solution when the moduli are pairwise coprime.
For instance, if you're looking for a number that leaves a remainder of 1 when divided by 3, and 2 when divided by 5, CRT helps you find it (the number is 7, and all numbers congruent to 7 mod 15). Essential in cryptography and computer science, CRT simplifies problems involving modular arithmetic.
- Definition: Finds a number satisfying a system of congruences.
- Use Cases: Cryptography, coding theory, computer algorithms.
- Formula: Involves modular inverses and product of moduli.