Solve System of Congruences Using CRT

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In summary, the Chinese Remainder Theorem (CRT) is a mathematical concept that simplifies the process of solving systems of congruences by breaking it down into smaller, simpler congruences with relatively prime moduli. The steps involved in solving a system of congruences using CRT are identifying the moduli, applying the CRT formula to each congruence, and then combining the solutions to get the final solution. CRT has various applications in mathematics, computer science, and cryptography, but it can only be applied to systems of congruences with pairwise relatively prime moduli.
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needmathshelp
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Hi, i know how to do the CRT but I'm kind at a loss with this one. Can anyone help me please?

Explain why the Chinese Remainder Theorem does not apply directly to the following
system.
x = 5 mod 6
x = 17 mod 21

Determine an equivalent system of equations to which the CRT applies and hence solve this
system of congruences.

Cheers.
 
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What does the Chinese remainder theorem state? In particular, what conditions must be satisfied before it can be applied? Which condition or conditions aren't satisfied by this example?
 

What is the Chinese Remainder Theorem (CRT)?

The Chinese Remainder Theorem is a mathematical concept that states that if we have a system of congruences with pairwise relatively prime moduli, then there exists a unique solution within a certain range of values.

How does CRT help in solving systems of congruences?

CRT simplifies the process of solving systems of congruences by breaking it down into smaller, simpler congruences with relatively prime moduli. This makes it easier to find the solution to the overall system of congruences.

What are the steps involved in solving a system of congruences using CRT?

The steps involved in solving a system of congruences using CRT are:

  1. Identify the moduli in the system of congruences and ensure they are pairwise relatively prime.
  2. Apply the Chinese Remainder Theorem formula to find the solution to each congruence individually.
  3. Combine the solutions using the Chinese Remainder Theorem formula to get the final solution to the system of congruences.

Can CRT be applied to any system of congruences?

No, CRT can only be applied to systems of congruences with pairwise relatively prime moduli. If the moduli are not relatively prime, then CRT cannot be used to find the solution.

What are the applications of CRT in real life?

CRT has various applications in mathematics, computer science, and cryptography. It is used in solving linear congruences and systems of linear congruences, as well as in data encryption and decryption algorithms.

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