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Chinese Remainder Theorem

  This theorem, as the name suggests, is one of those known to Chinese mathematicians a long time before the West. It is closely related to the work of Euclid (c. 295 BCE) on number theory, and is usually proved with the use of Euclid\'s algorithm, though Euclid himself did not know the result. At first sight the theorem seems unlikely and counter-intuitive. It states that given any finite set of numbers with no factors in common (for instance, 3, 5, and 17) and another set of remainders (say, 2, 4 and 3) it is possible to find a number which leaves the first remainder when divided by the first number (here, remainder 2 when divided by 3), the second remainder when divided by the second number, and so on (here, such a number is 224). In fact, there are infinitely many such numbers, since adding the product of the original set of numbers (3 × 5 × 17 = 255) any number of times will give another number which works. SMcL  



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