Solving Systems of Congruences when mods not pairwise relatively prime

In summary, when mods are not pairwise relatively prime, it means that they do not share any common factors other than 1. To solve systems of congruences in this case, the Extended Euclidean Algorithm can be used to find the inverse of each modulus, and then the Chinese Remainder Theorem can be applied to find the unique solution. However, there are challenges in solving these systems, such as finding the inverses and ensuring consistency. Special cases to consider include when one or more congruences have a modulus of 1 or when they have a common factor. This concept has practical applications in cryptography, error-correcting codes, and scheduling.
  • #1
CantorSet
44
0
Hi folks,

The CRT says there's a unique solution to the system of congruences

[itex] x = a [/itex] (mod m)
[itex] x = b [/itex] (mod n)
[itex] x = c [/itex] (mod p)

in (mod mnp) when [itex] m, n, p [/itex] are pairwise relatively prime. But what if [itex] m, n, p [/itex] are NOT pairwise relatively prime. Is there a systematic way to solve these cases?
 
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  • #2
The system may not have a solution if the moduli are not pairwise coprime.We can, of course,solve two equations at a time modulo the lcm & try to patch up the solutions... I don't know how to answer this best.
 

FAQ: Solving Systems of Congruences when mods not pairwise relatively prime

What does it mean for mods to not be pairwise relatively prime?

When mods are not pairwise relatively prime, it means that they do not share any common factors other than 1. This can make solving systems of congruences more complicated, as the Chinese Remainder Theorem cannot be directly applied.

How do you solve systems of congruences when mods are not pairwise relatively prime?

To solve systems of congruences when mods are not pairwise relatively prime, you can use the Extended Euclidean Algorithm to find the inverse of each modulus. Then, you can use the Chinese Remainder Theorem to combine the congruences and find the unique solution.

What are some challenges when solving systems of congruences with non-pairwise relatively prime mods?

Some challenges when solving systems of congruences with non-pairwise relatively prime mods include finding the inverse of each modulus, as well as ensuring that the congruences are all consistent and solvable. Additionally, the resulting solution may be larger than expected, as it will be a multiple of the product of the mods.

Are there any special cases to consider when solving systems of congruences with non-pairwise relatively prime mods?

Yes, there are a few special cases to consider when solving systems of congruences with non-pairwise relatively prime mods. One is when one or more of the congruences have a modulus of 1, which means that the solution will be the same as the corresponding residue. Another is when the congruences have a common factor, in which case the solution will have infinitely many solutions.

How is solving systems of congruences with non-pairwise relatively prime mods useful in real-world applications?

Solving systems of congruences with non-pairwise relatively prime mods is useful in cryptography, as it allows for the efficient decryption of encrypted messages. It is also used in error-correcting codes and in finding solutions to linear equations with modular arithmetic. Additionally, it has applications in scheduling, where congruences can represent different periods of time and their intersections.

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