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Chinese Remainder Theorem


Let r and s be positive integers which are relatively prime and let a and b be any two integers. Then there is an integer N such that

 N=a (mod r)
(1)

and

 N=b (mod s).
(2)

Moreover, N is uniquely determined modulo rs. An equivalent statement is that if (r,s)=1, then every pair of residue classes modulo r and s corresponds to a simple residue class modulo rs.

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

 x=a_i (mod m_i)
(3)

for i=1, ..., r and for which the m_i are pairwise relatively prime, the solution of the set of congruences is

 x=a_1b_1M/(m_1)+...+a_rb_rM/(m_r) (mod M),
(4)

where

 M=m_1m_2...m_r
(5)

and the b_i are determined from

 b_iM/(m_i)=1 (mod m_i).
(6)

See also

Congruence, Congruence Equation, Linear Congruence Equation

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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. https://mathworld.wolfram.com/ChineseRemainderTheorem.html

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