Chinese Remainder Theorem
Formally stated, the Chinese Remainder Theorem is as follows:
Let be relatively prime to . Then each residue class mod is equal to the intersection of a unique residue class mod and a unique residue class mod , and the intersection of each residue class mod with a residue class mod is a residue class mod .
Suppose you wish to find the least number which leaves a remainder of:
such that , , ... are all relatively prime. Let , and . Now if the numbers satisfy:
for every , then a solution for is:
If , then and differ by a multiple of , so and . This is the first part of the theorem. The converse follows because and must differ by a multiple of and , and . This is the second part of the theorem.
Much like the Fundamental Theorem of Arithmetic, many people seem to take this theorem for granted before they consciously turn their attention to it. Its ubiquity derives from the fact that many results can be easily proven mod (a power of a prime), and can then be generalized to mod using the Chinese Remainder Theorem. For instance, Fermat's Little Theorem may be generalized to the Fermat-Euler Theorem in this manner.
General Case: the proof of the general case follows by induction to the above result (k-1) times.
Extended version of the theorem
Suppose one tried to divide a group of fish into , and parts instead and found , and fish left over, respectively. Any number with remainder mod must be odd and any number with remainder mod must be even. Thus, the number of objects must be odd and even simultaneously, which is a contradiction. Thus, there must be restrictions on the numbers to ensure that at least one solution exists. It follows that:
The solution exists if and only if for all where stands for the greatest common divisor. Moreover, in the case when the problem is solvable, any two solutions differ by some common multiple of . (the extended version).
- Here is an AoPS thread in which the Chinese Remainder Theorem is discussed and implemented.