Theorem 1.7.35 (Chinese remainder theorem, general version) Let be pairwise relatively prime integers. Let . Then

(1.7)

has a simultaneous solution . Furthermore, if are two solutions to (1.7) then .

This follows from the case proven above using mathematical induction. The details are left as an exercise. We give a different proof below.

proof: As runs over all integers , the -tuples

form a collection of distinct -tuples in . (Exercise: show why they are distinct.) On the other hand, there are distinct, -tuples with . Therefore, each -tuple must equal one of the , for .