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)
|
