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Wolfram Web Resources » 13,585 entries Last updated: Sun Apr 3 2016 Created, developed, and nurtured by Eric Weisstein at Wolfram Research | Number Theory > Congruences > Interactive Entries > Interactive Demonstrations > # Chinese Remainder Theorem Let and be positive integers which are relatively prime and let and be any two integers. Then there is an integer such that
| (1)
and
| (2)
Moreover, is uniquely determined modulo . An equivalent statement is that if , then every pair of residue classes modulo and corresponds to a simple residue class modulo . The Chinese remainder theorem is implemented in the Wolfram Language as `ChineseRemainder`[a1, a2, ...m1, m2, ...]. The Chinese remainder theorem is also implemented indirectly using `Reduce` in with a domain specification of `Integers`. The theorem can also be generalized as follows. Given a set of simultaneous congruences
| (3)
for , ..., and for which the are pairwise relatively prime, the solution of the set of congruences is
| (4)
where
| (5)
and the are determined from
| (6)
SEE ALSO: Congruence, Congruence Equation, Linear Congruence Equation REFERENCES: Flannery, S. and Flannery, D. In Code: A Mathematical Journey. London: Profile Books, pp. 123-125, 2000. Ireland, K. and Rosen, M. "The Chinese Remainder Theorem." §3.4 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 34-38, 1990. Séroul, R. "The Chinese Remainder Theorem." §2.6 in Programming for Mathematicians. Berlin: Springer-Verlag, pp. 12-14, 2000. Uspensky, J. V. and Heaslet, M. A. Elementary Number Theory. New York: McGraw-Hill, pp. 189-191, 1939. Wagon, S. "The Chinese Remainder Theorem." §8.4 in Mathematica in Action. New York: W. H. Freeman, pp. 260-263, 1991. CITE THIS AS: Weisstein, Eric W. "Chinese Remainder Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ChineseRemainderTheorem.html # Wolfram Web Resources
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