Transcription of The Chinese Remainder Theorem
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The Chinese Remainder Theorem
The Chinese Remainder TheoremWe now know how to solve a single linear congruence. In this lecture weconsider how to solve systems ofsimultaneouslinear solve the system 2x 5 (mod 7); 3x 4 (mod 8) of twolinear congruences (in one variablex). Multiply the first congruence by 2 1mod 7 = 4 to get 4 2x 4 5 (mod 7). This simplifies tox 6 (mod 7),sox= [6]7orx= 6 + 7t, wheret substitute forxin the second congruence: 3(6 + 7t) 4 (mod 8). Thissimplifies to 5t 2 (mod 8), which we solve by multiplying both sides by5 1mod 8 = 5 to obtaint 2 (mod 8).
The Chinese Remainder Theorem We now know how to solve a single linear congruence. In this lecture we consider how to solve systems of simultaneous linear congruences.
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