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Chinese Remainder Calculator

The Remainders in China Theorem refers to a system that contains congruences, and multiple simultaneous modular systems. The problem lies in calculating the number of elements that remain and how to solve it. 

chinese remainder calculator

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Table of contents

Introduction to the Chinese Remainder Theorem
Modulo and congruences
The Chinese remainder theorem 
Euclidean gcd algorithm
Bezout's identity

Introduction to the Chinese Remainder Theorem

Have you ever heard of the Chinese Remainder Theorem? It is a system that deals with congruences and simultaneous modular systems, and is used to calculate the number of elements that remain in the system and how to solve it.
In this article, we will delve into the concept of modulo and congruences, and how they relate to the Chinese Remainder Theorem. We will also discuss the Euclidean gcd algorithm and Bezout's identity, two important concepts that are closely related to the Chinese Remainder Theorem.
So, if you want to learn more about this fascinating area of mathematics, keep reading!
The Remainders in China Theorem refers to a system that contains congruences, and multiple simultaneous modular systems. The problem lies in calculating the number of elements that remain and how to solve it. 

Modulo and congruences

Are you familiar with the mathematics that deals with numbers? Apparently, is not. The big-headed mathematicians are able to find more and better ways to describe and analyze the objects from their wildest imaginations. Most often, there are no digits in the formulas they're proving.
Number Theory is the last of this sea of unrecognizable symbols. It is a mathematics field that only deals with integers. Numbers like 0, 1, 42, or 273 are examples. That's right, don't bother with fractions. It seems like a narrow field, doesn’t it? They must have found everything by now.
The remainder of an integer divided by b, which is the integer r between zero and b-1 that remains as the extra "undivided", or the one that would give a fraction from the operation, is called the "undivided". It must satisfy the equation, symbolically speaking.
a = k * b + r (k being some integer)
If we divide 17 by 5, then we know we can fit three 5-s into 17. And we'll still have 2 standing. The 2 represents the remainder. 
17 = 3 * 5 + 2

The Chinese remainder theorem 

This is a method that describes a set or system of congruences that describe an unknown variable. The statement will be revealed soon enough. For now, however, we'll just say that it solves such problems and provides a simple algorithm for doing so.

Euclidean gcd algorithm

The Euclidean-gcd algorithm allows you to find the greatest Common Divisor (sometimes referred to as the Greatest Common Factor) of two numbers.

Bezout's identity

Bezout’s identity (also known as Bezout’s lemma, not to be confused avec Bezout’s theorem which deals with dividing polynomials), is a small theorem that allows us to connect two numbers using their respective gcd.
John Cruz
Article author
John Cruz
John is a PhD student with a passion to mathematics and education. In his freetime John likes to go hiking and bicycling.
Chinese Remainder Calculator English
Published: Thu Mar 23 2023
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