Teorema Chinês (2 congruências)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
TCR
About this calculator
The Chinese Remainder Theorem calculator is a useful tool for solving systems of congruences, which are mathematical problems that involve finding a value x that satisfies multiple modular equations. These equations have the form x ≡ a₁ (mod m₁) and x ≡ a₂ (mod m₂), where a₁, a₂, m₁, and m₂ are integers.
The Chinese Remainder Theorem formula is used to solve this type of problem. It states that, if m₁ and m₂ are relatively prime, then there exists a unique value x that satisfies the modular equations. Furthermore, the value of x can be found using the formula x ≡ a₁m₂y₁ + a₂m₁y₂ (mod m₁m₂), where y₁ and y₂ are the inverses of m₂ and m₁, respectively.
The Chinese Remainder Theorem calculator is useful for solving problems in Number Theory, which are problems that involve properties and structures of the integers. It is especially useful for solving problems that involve congruences and modules.
Frequently asked questions
What is the Chinese Remainder Theorem?
The Chinese Remainder Theorem is a theorem in Number Theory that provides a necessary and sufficient condition for a system of congruences to have a unique solution.
When to use the Chinese Remainder Theorem calculator?
Use the Chinese Remainder Theorem calculator when you need to solve a system of congruences, i.e., find a value x that satisfies multiple modular equations.
How does the Chinese Remainder Theorem calculator work?
The Chinese Remainder Theorem calculator uses the theorem's formula to find the value of x that satisfies the modular equations. It calculates the inverses of m₂ and m₁, and then uses the formula x ≡ a₁m₂y₁ + a₂m₁y₂ (mod m₁m₂) to find the value of x.
What is important to know about the Chinese Remainder Theorem calculator?
It is important to know that the Chinese Remainder Theorem calculator only works if m₁ and m₂ are relatively prime. Additionally, the value of x found by the calculator is unique.