Chinese remainder theorem

Chinese remainder theorem, ancient theorem that gives the conditions necessary for multiple equations to have a simultaneous integer solution. The theorem has its origin in the work of the 3rd-century-ad Chinese mathematician Sun Zi, although the complete theorem was first given in 1247 by Qin Jiushao.

The Chinese remainder theorem addresses the following type of problem. One is asked to find a number that leaves a remainder of 0 when divided by 5, remainder 6 when divided by 7, and remainder 10 when divided by 12. The simplest solution is 370. Note that this solution is not unique, since any multiple of 5 × 7 × 12 (= 420) can be added to it and the result will still solve the problem.

The theorem can be expressed in modern general terms using congruence notation. (For an explanation of congruence, see modular arithmetic.) Let n₁, n₂, …, n_k be integers that are greater than one and pairwise relatively prime (that is, the only common factor between any two of them is 1), and let a₁, a₂, …, a_k be any integers. Then there exists an integer solution a such that a ≡ a_i (mod n_i) for each i = 1, 2, …, k. Furthermore, for any other integer b that satisfies all the congruences, b ≡ a (mod N) where N = n₁n₂⋯n_k. The theorem also gives a formula for finding a solution. Note that in the example above, 5, 7, and 12 (n₁, n₂, and n₃ in congruence notation) are relatively prime. There is not necessarily any solution to such a system of equations when the moduli are not pairwise relatively prime.