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DOCUMENTATION CENTER SEARCH
New to
Mathematica
?
Find your learning path
»
Mathematica
>
Mathematics and Algorithms
>
Number Theory
>
Diophantine Equations
>
ChineseRemainder
>
BUILT-IN MATHEMATICA SYMBOL
Integer and Number Theoretic Functions
Tutorials »
|
Reduce
FindInstance
GCD
See Also »
|
Diophantine Equations
Number Theoretic Functions
Number Theory
New in 6.0: Mathematics & Algorithms
New in 6.0: Number Theory & Integer Functions
More About »
ChineseRemainder
ChineseRemainder
gives the smallest non-negative
x
that satisfies all the integer congruences
x
mod
=
mod
.
MORE INFORMATION
If no solution for
x
exists,
ChineseRemainder
returns unevaluated.
If all
, then the result satisfies
x
mod
=
.
EXAMPLES
CLOSE ALL
Basic Examples
(2)
The smallest positive integer
x
that satisfies
and
:
Find the smallest positive integer giving remainder
when divided by
:
The smallest positive integer
x
that satisfies
and
:
In[1]:=
Out[1]=
Find the smallest positive integer giving remainder
when divided by
:
In[1]:=
Out[1]=
Applications
(3)
Database encryption and decryption:
Key generation:
Encrypted data:
Decryption:
Define a residue number system:
Numbers and their representation in a residue system:
Multiplying and recovering in the residue system:
Adding and recovering:
Modular computation of a determinant:
Modular determinants:
Recover result:
Shift residue to be symmetric:
Properties & Relations
(1)
Solve congruential equations using
Reduce
or
FindInstance
:
Possible Issues
(1)
Not all congruential equations have a solution:
A solution exists when
Mod
[
r
i
,
GCD
[
m
1
,
m
2
, ...]]==
Mod
[
r
j
,
GCD
[
m
1
,
m
2
, ...]]
:
SEE ALSO
Reduce
FindInstance
GCD
TUTORIALS
Integer and Number Theoretic Functions
MORE ABOUT
Diophantine Equations
Number Theoretic Functions
Number Theory
New in 6.0: Mathematics & Algorithms
New in 6.0: Number Theory & Integer Functions
New in 6