# ChineseRemainder

ChineseRemainder[{r1,r2,},{m1,m2,}]

gives the smallest with that satisfies all the integer congruences .

ChineseRemainder[{r1,r2,},{m1,m2,},d]

gives the smallest with that satisfies all the integer congruences .

# Details • If no solution for exists, ChineseRemainder returns unevaluated.
• If all 0ri<mi, then the result satisfies .
• ChineseRemainder[{r1,r2,},{m1,m2,}] gives a solution with .
• ChineseRemainder[{r1,r2,},{m1,m2,},d] gives a solution with .

# Examples

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## Basic Examples(2)

The smallest positive integer that satisfies and :

Find the smallest positive integer giving remainders when divided by :

## 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:

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[ri,GCD[m1,m2,]]==Mod[rj,GCD[m1,m2,]]: