Problem Chinese remainder Theorem

MHB
Thread starter Julio1
Start date Oct 25, 2015
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Remainder Remainder theorem Theorem

In summary, the Chinese remainder theorem is a mathematical theorem used for finding a number that leaves specified remainders when divided by a set of given numbers. The problem Chinese remainder theorem is a variant that deals with finding the smallest number that satisfies a set of congruences. This theorem has applications in number theory and cryptography, and is solved by writing the congruences in a specific form and using the Chinese remainder theorem algorithm. However, it has limitations such as only being applicable to pairwise coprime moduli and becoming computationally intensive for large numbers. It also only applies to systems of linear congruences and cannot be used for other types of equations.

Oct 25, 2015

Julio1

Find the set of solutions $x=x(r,s,t)$ such that $(r+2\mathbb{N})\cap (s+3\mathbb{N})\cap (t+5\mathbb{N})=x+n\mathbb{N}.$

Hello MHB :). Any hints for the problem?

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Oct 26, 2015

Euge

Gold Member

MHB

POTW Director

Hi Julio,

Could you specify what $n$ is? Is it $30$?

Nov 22, 2015

Julio1

Euge said:

Hi Julio,

Could you specify what $n$ is? Is it $30$?

Hi Euge :).

Yes, $n=\text{lcm}(2,3,5)=30.$

1. What is the Chinese remainder theorem?

The Chinese remainder theorem is a mathematical theorem that describes a procedure for finding a number that, when divided by a set of given numbers, leaves specified remainders. In other words, it is a way to solve a system of congruences.

2. What is the problem Chinese remainder theorem?

The problem Chinese remainder theorem is a variant of the Chinese remainder theorem that deals with finding the smallest number that satisfies a given set of congruences. This number is also known as the "least common multiple" or "smallest solution" of the congruences.

3. What are some applications of the Chinese remainder theorem?

The Chinese remainder theorem has many applications in number theory and cryptography. It is used to solve systems of linear congruences, to find primitive roots modulo a number, and to construct secret sharing schemes. It is also used in the RSA algorithm for public-key encryption.

4. How do you solve a problem using the Chinese remainder theorem?

To solve a problem using the Chinese remainder theorem, you need to first write the given congruences in the form x ≡ a (mod m). Then, use the Chinese remainder theorem algorithm to find the smallest solution x. This involves calculating the least common multiple of the moduli and using modular arithmetic to find the value of x.

5. Are there any limitations or drawbacks to using the Chinese remainder theorem?

One limitation of the Chinese remainder theorem is that it can only be used when the moduli are pairwise coprime (meaning they have no common factors). Additionally, the algorithm can become computationally intensive when dealing with large numbers, making it impractical for certain applications. Finally, the theorem only applies to systems of linear congruences, so it cannot be used for solving other types of equations.