chinese remainder theorem
Chinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra. It was first published in the 3rd to 5th centuries by Chinese mathematician Sun Tzu.
In its basic form, the Chinese remainder theorem will determine a number n that when divided by some given divisors leaves given remainders. For example, what is the lowest number n that when divided by 3 leaves a remainder of 2, when divided by 5 leaves a remainder of 3, and when divided by 7 leaves a remainder of 2?
Theorem statement
The original form of the theorem, contained in the 5th-century book Sunzi's Mathematical Classic (孫子算經) by the Chinese mathematician Sun Tzu and later generalized with a complete solution called Dayanshu (大衍術) in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections (數書九章, Shushu Jiuzhang), is a statement about simultaneous congruences.
Suppose n1, n2, …, nk are positive integers that are pairwise coprime. Then, for any given sequence of integers a1,a2, …, ak, there exists an integer x solving the following system of simultaneous congruences.
Furthermore, all solutions x of this system are congruent modulo the product, N = n1n2…nk.
Hence for all , if and only if .
Sometimes, the simultaneous congruences can be solved even if the ni's are not pairwise coprime. A solution x exists if and only if:
All solutions x are then congruent modulo the least common multiple of the ni.
Sun Tzu's work contains neither a proof nor a full algorithm. What amounts to an algorithm for solving this problem was described by Aryabhata (6th century; see Kak 1986). Special cases of the Chinese remainder theorem were also known to Brahmagupta (7th century), and appear in Fibonacci's Liber Abaci (1202).
A modern restatement of the theorem in algebraic language is that for a positive integer with prime factorization we have the isomorphism between a ring and the direct product of its prime power parts:
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra. It was first published in the 3rd to 5th centuries by Chinese mathematician Sun Tzu.
In its basic form, the Chinese remainder theorem will determine a number n that when divided by some given divisors leaves given remainders. For example, what is the lowest number n that when divided by 3 leaves a remainder of 2, when divided by 5 leaves a remainder of 3, and when divided by 7 leaves a remainder of 2?
Theorem statement
The original form of the theorem, contained in the 5th-century book Sunzi's Mathematical Classic (孫子算經) by the Chinese mathematician Sun Tzu and later generalized with a complete solution called Dayanshu (大衍術) in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections (數書九章, Shushu Jiuzhang), is a statement about simultaneous congruences.
Suppose n1, n2, …, nk are positive integers that are pairwise coprime. Then, for any given sequence of integers a1,a2, …, ak, there exists an integer x solving the following system of simultaneous congruences.
Furthermore, all solutions x of this system are congruent modulo the product, N = n1n2…nk.
Hence for all , if and only if .
Sometimes, the simultaneous congruences can be solved even if the ni's are not pairwise coprime. A solution x exists if and only if:
All solutions x are then congruent modulo the least common multiple of the ni.
Sun Tzu's work contains neither a proof nor a full algorithm. What amounts to an algorithm for solving this problem was described by Aryabhata (6th century; see Kak 1986). Special cases of the Chinese remainder theorem were also known to Brahmagupta (7th century), and appear in Fibonacci's Liber Abaci (1202).
A modern restatement of the theorem in algebraic language is that for a positive integer with prime factorization we have the isomorphism between a ring and the direct product of its prime power parts: