Statement of Chinese Remainder Theorem

Theorem states that a system of simultaneous congruences, defined by pairwise coprime positive integers (n1, n2,…., nk ) and arbitrary integers (a1, a2,…., ak )

x ≡ a1 (mod n1)

x ≡ a2 (mod n2)

x ≡ ak (mod nk)

has a unique solution modulo (N = n1 n2 …. nk)

Chinese Remainder Theorem

Chinese Remainder Theorem is a mathematical principle that solves systems of modular equations by finding a unique solution from the remainder of the division. It is used in cryptography and computer science for efficient computation.

In this article, we will learn the meaning and definition of the Chinese Remainder Theorem, the history of the Chinese Remainder Theorem, the statement and proof of the Chinese Remainder Theorem and applications of the Chinese Remainder Theorem.

Table of Content

  • What is The Chinese Statement Theorem?
  • History of Sun Zi
  • Statement of Chinese Remainder Theorem
  • Chinese Remainder Theorem Proof
  • Application of the Chinese Remainder Theorem

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What is Chinese Remainder Theorem?

According to the Chinese Remainder Theorem, if in a given set of equations, each equation has a different number (say, n1, n2, …, nk), and these numbers are all relatively prime to each other, and also have some numbers (say, a1, a2,…, ak), then there is a unique solution (let’s call it x) that satisfies all the equations at once....

Statement of Chinese Remainder Theorem

Theorem states that a system of simultaneous congruences, defined by pairwise coprime positive integers (n1, n2,…., nk ) and arbitrary integers (a1, a2,…., ak )...

Chinese Remainder Theorem Proof

To proof Chinese remainder theorem, let us find an integer (x) that satisfies: x ≡ ai (mod ni) for each (i = 1, 2,…., k)....

Application of the Chinese Remainder Theorem

Applications of the Chinese remainder theorem are as follows:...

Example on Chinese Remainder Theorem

Example 1: Suppose a certain number leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 5, and a remainder of 2 when divided by 7. Find the smallest positive integer that satisfies these conditions using the Chinese Remainder Theorem....

Practice Questions

Q1. Solve the system of congruences:...

FAQs on Chinese Remainder Theorem

Why is it Called Chinese Remainder Theorem?...