# 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,]]:

Wolfram Research (2007), ChineseRemainder, Wolfram Language function, https://reference.wolfram.com/language/ref/ChineseRemainder.html (updated 2016).

#### Text

Wolfram Research (2007), ChineseRemainder, Wolfram Language function, https://reference.wolfram.com/language/ref/ChineseRemainder.html (updated 2016).

#### CMS

Wolfram Language. 2007. "ChineseRemainder." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/ChineseRemainder.html.

#### APA

Wolfram Language. (2007). ChineseRemainder. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ChineseRemainder.html

#### BibTeX

@misc{reference.wolfram_2024_chineseremainder, author="Wolfram Research", title="{ChineseRemainder}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/ChineseRemainder.html}", note=[Accessed: 26-May-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_chineseremainder, organization={Wolfram Research}, title={ChineseRemainder}, year={2016}, url={https://reference.wolfram.com/language/ref/ChineseRemainder.html}, note=[Accessed: 26-May-2024 ]}