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The Chinese Remainder Theorem

The Chinese Remainder Theorem Chinese Remainder Theorem : If m1, m2, .., mk are pairwise relatively prime positive integers, and if a1, a2, .., ak are any integers, then the simultaneous congruences x a1 (mod m1), x a2 (mod m2), .., x ak (mod mk) have a solution, and the solution is unique modulo m, where m = m1m2 mk. Proof that a solution exists: To keep the notation simpler, we will assume k = 4. Note the proof is constructive, , it shows us how to actually construct a solution. Our simultaneous congruences are x a1 (mod m1), x a2 (mod m2), x a3 (mod m3), x a4 (mod m4). Our goal is to find integers w1, w2, w3, w4 such that: value mod m1 value mod m2 value mod m3 value mod m4w1 1 0 0 0 w2 0 1 0 0 w3 0 0 1 0 w4 0 0 0 1 Once we have found w1, w2, w3, w4, it is easy to construct x.

The Chinese Remainder Theorem Chinese Remainder Theorem: If m 1, m 2, .., m k are pairwise relatively prime positive integers, and if a 1, a 2, .., a

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