The Chinese remainder theorem states that if the remainders of an integer's division by several pairwise coprime integers are known, the remainder of its division by the product of those integers can be uniquely determined. The earliest known special case appears in the Sunzi Suanjing, and Qin Jiushao gave the complete general solution in his 1247 treatise. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
StatementGiven several pairwise coprime divisors and the remainder an integer leaves under each one, there is exactly one remainder that integer can leave under the product of all those divisors, and it is uniquely determined by the individual remainders. 1 Classification
Statement Form Connections
Associated With
The theorem's statement is phrased entirely in remainders of integer division by a modulus, the operation modular arithmetic defines.
Additional Source Modular Arithmetic (Wikipedia)Lead section
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Chinese Remainder Theorem (Wikipedia)
Proved By
Source Chinese Remainder Theorem (Wikipedia)
Sources
1. Chinese Remainder Theorem (Wikipedia)
Wikimedia Foundationintroduction
one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime
History section
Qin Jiushao provided a complete general solution in his 1247 work Mathematical Treatise in Nine Sections
Lead section, statement-form reference
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime (no two divisors share a common factor other than 1).
View the Source Sunzi Suanjing (Wikipedia)
Wikimedia Foundationsignificance sectionQuote, significance section
Chapter 3 contains the earliest example of the Chinese remainder theorem, a key tool to understanding and resolving Diophantine equations.
Modular Arithmetic (Wikipedia)
Wikimedia FoundationAssociated With: Modular Arithmetic, Lead sectionQuote, Associated With: Modular Arithmetic, Lead section
modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when reaching or exceeding a certain value, called the modulus
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