Difference between revisions of "Chinese Remainder Theorem"
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Revision as of 17:00, 25 December 2007
The Chinese Remainder Theorem is a number theoretic result.
History
When the ancient Chinese army was at its peak, it was said that the emperor developed a way to count the soldiers. Rather than count his hordes one man at a time, he developed a system. He would have his soldiers divide into groups of 3, and count the men who didn't have a group. He would then do groups of 5, etc. Eventually, using these remainders, he could calculate the size of his army using a technique we now call the Chinese Remainder Theorem.
Proof
If , then
and
clearly 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.
Applicability
Much like the Fundamental Theorem of Arithmetic, many people seem to take this theorem for granted before they consciously turn their attention to it. It 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 intance, Fermat's Little Theorem may be generalized to the Fermat-Euler Theorem in this manner. Its application in problem-solving is similar.
General Case: the proof of the general case follows by repeatedly applying to above result (k-1) times.
Extended version of the theorem
Suppose one tried to divide a group of objects into ,
and
parts instead and found
,
and
fish left over, respectively. One can show this to be impossible, because any number giving remainder
modulo
must be odd and any number giving remainder
modulo
must be even. Thus, the number of objects must be odd and even simultaneously, which is absurd. Thus, there must be some restrictions on the numbers
to ensure that at least one solution exists. It turns out (and this is the extended version of the theorem) 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
.
Discussion
- Here is an AoPS thread in which the Chinese Remainder Theorem is discussed and implemented.