# China Remainder Theorem

Source: Internet
Author: User

China Remainder Theorem

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China Remainder Theorem(Chinese Remainder Theorem,Chinese Remainder Theorem).Han Xin dianbing","Sun Tzu's Theorem","Ghost Valley computing","Partition Calculation","Tube Cutting","Wang Qin's secret","UnknownThe name is an important proposition in number theory.

 Directory[Hide] 1 Unknown 2 Format description 3 Colick 4 Recent world exchange circle and promotion 5 Bibliography

 Unknown

In the famous ancient Chinese Mathematical book Sun Tzu's computing Sutra, there is a title called "unknown things". The original Article is as follows:

I don't know the number of things, but the number of three is two, the number of five is three, and the number of seven is two. Ry?

That is, an integer is divided by three integers, divided by five integers, and divided by seven integers.

Three people walk 70 Greek, five trees plum blossom one, seven son reunion is half a month, except five will know

This solution is actually, first, we use the derivative technique invented by Qin Jiuyi to find the smallest 70 in the multiples of the minimum public multiples of 5 and 7 35 divided by the remainder of 3 (this is called the reciprocal of 35 to 3 in number theory ), the minimum public multiples of 3 and 7, 21, and 5, respectively, 15, and 15, respectively, are opposite to the reciprocal of 7. Then

233 is one of the possible solutions. The addition and subtraction of 3, 5, and 7 is several times the minimum public multiple of 105, so the minimum solution is 233 divided by the remainder of 105 23.

Note: This solution is not the simplest, because in fact 35 is in line with the characteristics except 3 + 2, so the half solution is: the half solution plus a positive integer multiple of 105 is the solution.

 Format description

If the above solution is generalized to the general situation, a constructive Proof of the Chinese Remainder Theorem is obtained.

Generally,China Remainder TheoremIt refers to any integer if there are two-byte integers:A1,A2 ,...AN, The following simultaneous equations have a public solution to the modulus:

 Colick

To facilitate expression, use a common function for any positive integerεI,JIt is called the Kronecker. Definition:

With this symbol, the process of solving the above general equations of the same equations can be given, which is completed in two steps.

• For each, first obtain a positive integerBISatisfied, that is, what you wantBIConditions met:, but is divisible by each. The solution is as follows: Note: we can see that, according to the conditionsRIAndMIAlso, there are integers.CIAndDIMakeRICI+MIDI= 1.BI=RICI, CorrespondsMJApparently DivisionBIAnd. This indicatesBIThat is, the request.
• For the aforementionedBI, Order, then, this descriptionX0It is a solution of the same equations above, so all solutions can be expressed as, where n can retrieve all integers.
 Recent world exchange circle and promotion

An exchange ring with a unit element is the ideal of a ring. At that time, a typical ring is homogeneous, where the ring structure is given by ing.

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