Math 2.0

Number theory Congruences Theorem

Chinese remainder theorem

Builds on Congruence modulo nCoprime integers

Statement

Let n1,…,nkn_1, \dots, n_k be positive integers that are pairwise coprime, and let N=n1n2⋯nkN = n_1 n_2 \cdots n_k. Let a1,…,aka_1, \dots, a_k be integers. Then there is an integer xx such that x≡ai(modni)x \equiv a_i \pmod{n_i} for each ii. Two such integers are congruent modulo NN.

Remarks

In the language of rings, the map ℤ/Nℤ→ℤ/n1ℤ×⋯×ℤ/nkℤ\mathbb{Z}/N\mathbb{Z} \to \mathbb{Z}/n_1\mathbb{Z} \times \cdots \times \mathbb{Z}/n_k\mathbb{Z}, x‾↦(x‾,…,x‾)\overline{x} \mapsto (\overline{x}, \dots, \overline{x}), is a ring isomorphism. For two moduli mm and nn, write mu+nv=1m u + n v = 1 with Bézout's identity. Then x=a2mu+a1nvx = a_2 m u + a_1 n v solves x≡a1(modm)x \equiv a_1 \pmod m and x≡a2(modn)x \equiv a_2 \pmod n. The Chinese remainder theorem for non-coprime moduli removes the hypothesis of coprimality.

History

The Chinese book Sunzi Suanjing, from the 5th century, contains the oldest known example of the problem. Qin Jiushao gave a complete method in 1247.

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Divisibility, definition, no evidence levelDivisibilitydefinitionCongruence modulo n, definition, no evidence levelCongruence modulo ndefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionCoprime integers, definition, no evidence levelCoprime integersdefinitionChinese remainder theorem, statement, no evidence levelChinese remainder the…statement
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  1. Divisibility definition
  2. Congruence modulo n definition
  3. Greatest common divisor definition
  4. Coprime integers definition
  5. Chinese remainder theorem statement

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Further reading

  1. Web https://en.wikipedia.org/wiki/Chinese_remainder_theorem, Statement
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Kindstatement, theorem
WritingW1 stated
MSC11A07
StatusThis copy of the site has no build data. Run m2 status 0FNN9B4W64.
UsesCongruence modulo n, Coprime integers
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