Chinese remainder theorem

Suppose we have a set of $n$ congruences of the form

\begin{array}[]{ccc}x&\equiv&a_{1}\pmod{p_{1}}\\ x&\equiv&a_{2}\pmod{p_{2}}\\ &\vdots&\\ x&\equiv&a_{n}\pmod{p_{n}}\\ \end{array}

where $p_{1},p_{2},\dots,p_{n}$ are relatively prime. Let

P=\prod_{i=1}^{n}p_{i}

and, for all $i\in\mathbb{N}$ ( $1\leq i\leq n$ ), let $y_{i}$ be an integer that satisfies

y_{i}\cdot\frac{P}{p_{i}}\equiv 1\pmod{p_{i}}

Then one solution of these congruences is

x_{0}=\sum_{i=1}^{n}a_{i}y_{i}\cdot\frac{P}{p_{i}}

Any $x\in\mathbb{Z}$ satisfies the set of congruences if and only if it satisfies

x\equiv x_{0}\pmod{P}

The Chinese remainder theorem originated in the book “Sun Zi Suan Jing”, or Sun Tzu’s Arithmetic Classic, by the Chinese mathematician Sun Zi, or Sun Tzu, who also wrote “Sun Zi Bing Fa”, or Sun Tzu’s The Art of War. The theorem is said to have been used to count the size of the ancient Chinese armies (i.e., the soldiers would split into groups of 3, then 5, then 7, etc, and the “leftover” soldiers from each grouping would be counted).

Title	Chinese remainder theorem
Canonical name	ChineseRemainderTheorem
Date of creation	2013-03-22 11:58:58
Last modified on	2013-03-22 11:58:58
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	13
Author	CWoo (3771)
Entry type	Theorem
Classification	msc 11D79
Related topic	ChineseRemainderTheoremInTermsOfDivisorTheory
Related topic	GodelsBetaFunction