LYCOS RETRIEVER 
Chinese Remainder Theorem: Numbers
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Chinese Remainder Theorem (CRT) The following problem was posed by Sunzi [Sun Tsu] (4th century AD) in the book Sunzi Suanjing: There are certain things whose number is unknown. Repeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What will be the number? The answer is hidden in the following song: Oystein Ore mentions another puzzle with a dramatic element from Brahma-Sphuta-Siddhanta (Brahma's Correct System) by Brahmagupta (born 598 AD): An old woman goes to market and a horse steps on her basket and crashes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left.
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The Chinese Remainder Theorem has been credited to Sun Tsu Suan-Ching, from the 4th century AD. Is it recorded as the following: There are certain things whose number is unknown. Repeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What will be the number? This problem is explained, solved and extended with a similar problem. It is a nice problem concerning least common multiples, greated common divisors and modular arithemetic.
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Example: The reason that the Chinese Remainder Theorem is named such is because Sun Tzu (544BC – 496BC) mentioned it in his Mathematical Manual. In essence, he asked for a number [X] such that
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Robert G. Wilson v, PhD ATP wrote: > Et al, > > Has any one out there programmed a NB for the Chinese Remainder > Theorem? Would very much appreciate the help on this and any other > Number Theory apps. > > Sincerely, > > Bob. Bob, You'll find ChineseRemainderThereom under Addons/StandardPackages/NumberTheory/NumberTheoryFunctions. ChineseRemainderTheorem was written by Stan Wagon and well documented in his book Mathematica in Action (Freeman / ISBN 0-7167-2229-1). You'll find a wide selection of other Number Theory functions in the same package, as well as packages on NT in the same directory.
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The chinese remainder theorem is important in actual implementations of cryptographic programs because it provides a nice fast way to do most operations on really large integers. Here's how it works: You can take your six biggest 32-bit integers that have no factors in common and multiply them together. This gives you a 185-bit number as a product. So, instead of using a positional or intuitive representation for numbers less than 185 bits, you can use their remainders modulo those six 32-bit integers as their representation.
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Problems of this kind are all examples of what universally became known as the Chinese Remainder Theorem. In mathematical parlance the problems can be stated as finding n, given its remainders of division by several numbers m1, m2, ..., mk:
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