The Chinese remainder theorem states that if positive integers
which are relatively prime and integers , then there
is an integer
(1)
and
(2)
Moreover, residue
classes modulo residue
class modulo
The Chinese remainder theorem is implemented in the Wolfram Language as ChineseRemainder a 1,
a 2, ...m 1,
m 2, ...Reduce Integers .
More generally, given simultaneous congruences
(3)
for relatively prime positive
integers , a solution is
(4)
where
(5)
and the
(6)
See also Congruence ,
Congruence
Equation ,
Linear Congruence Equation
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References Flannery, S. and Flannery, D. In Code: A Mathematical Journey. Ireland, K. and Rosen, M. "The Chinese Remainder Theorem."
§3.4 in A
Classical Introduction to Modern Number Theory, 2nd ed. Numberphile. "Chinese Remainder Theorem and
Cards." Aug. 8, 2018. https://www.youtube.com/watch?v=l9dXo5f3zDc . Séroul,
R. "The Chinese Remainder Theorem." §2.6 in Programming
for Mathematicians. Uspensky,
J. V. and Heaslet, M. A. Elementary
Number Theory. Wagon,
S. "The Chinese Remainder Theorem." §8.4 in Mathematica
in Action. Referenced
on Wolfram|Alpha Chinese Remainder Theorem
Cite this as:
Weisstein, Eric W. "Chinese Remainder Theorem."
From MathWorld https://mathworld.wolfram.com/ChineseRemainderTheorem.html
Subject classifications
and 
are positive integers
which are relatively prime and 
and 
are any two integers, then there
is an integer 
such that
is uniquely determined modulo 
. Equivalently, every pair of residue
classes modulo 
and 
corresponds to a unique residue
class modulo 
.
a1,
a2, ...
,
m1,
m2, ...
].
It is also implemented indirectly using Reduce
with a domain specification of Integers.
,
..., 
,
where the 
are pairwise relatively prime positive
integers, a solution is
are determined from
