The Chinese Remainder Theorem and Simultaneous Congruences

Abstract

One of the most useful and delightful entities in number theory is the Chinese Remainder Theorem (CRT). The CRT says that it is possible to reconstruct integers in a certain range from their residues modulo a set of coprime moduli. Thus, for example, the 10 integers in the range 0 to 9 can be reconstructed from their two residues modulo 2 and modulo 5 (the coprime factors of 10). Say the known residues of a decimal digit are r ₂ = 0 and r ₅ = 3; then the unknown digit is 8 (uniquely!).