Can $\varphi$ be used to prove the Chinese remainder theorem?

In summary, if the greatest common divisor of $b$ and $c$ is equal to $1$, then for any two natural numbers $r$ and $s$, there exists an integer $x$ between $1$ and $bc$ that is both in the sets $r+b\mathbb{N}$ and $s+c\mathbb{N}$. This can be shown by defining a function $\varphi$ that maps $x$ to the remainder when divided by $b$ and $c$ respectively, and proving that it is a bijection. This concept relates to the original statement as it provides a way to find an $x$ that satisfies the conditions.
  • #1
Julio1
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Show that if $\text{gcd}(b,c)=1$, then $\forall r,s\in \mathbb{N}, \exists x\in \{1,...,bc\}$ such that $x\in (r+b\mathbb{N})\cap (s+c\mathbb{N}).$

Hello :). Can define an function $\varphi: \{1,...,bc\}\to \mathbb{Z}/b\mathbb{Z}\times \mathbb{Z}/c\mathbb{Z}$ at follow $x\mapsto ([x]_b,[x]_c)$ all right? But what more can do? Thanks :).
 
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  • #2


Hello! Yes, that is a great start. Now, since $\text{gcd}(b,c)=1$, we know that $b$ and $c$ are relatively prime, which means that there exists integers $u$ and $v$ such that $ub+vc=1$. Using this information, we can show that $\varphi$ is a bijection. This means that for any $(r,s)\in \mathbb{Z}/b\mathbb{Z}\times \mathbb{Z}/c\mathbb{Z}$, there exists some $x\in \{1,...,bc\}$ such that $\varphi(x)=(r,s)$. Can you see how this relates to the original statement?
 

Related to Can $\varphi$ be used to prove the Chinese remainder theorem?

What is the Chinese Remainder Theorem?

The Chinese Remainder Theorem is a mathematical concept that deals with finding a solution to a system of congruences, where the divisors are pairwise relatively prime. It states that if there are several congruences with different moduli, it is possible to combine them to form a single congruence with a larger modulus, and still preserve the solutions. This theorem has important applications in number theory, cryptography, and computer science.

How is the Chinese Remainder Theorem used in cryptography?

The Chinese Remainder Theorem is used in cryptography to efficiently decrypt messages that have been encrypted using multiple keys. In this case, the multiple keys correspond to the different moduli in the congruences. By using the Chinese Remainder Theorem, the decryption process is faster and more efficient compared to using each key separately.

What are the limitations of the Chinese Remainder Theorem?

The Chinese Remainder Theorem has limitations when the divisors in the system of congruences are not pairwise relatively prime. In such cases, the theorem cannot be applied, and other methods must be used to find a solution. Additionally, the Chinese Remainder Theorem only applies to integer solutions and cannot be extended to real or complex numbers.

What are the applications of the Chinese Remainder Theorem?

The Chinese Remainder Theorem has various applications in number theory, cryptography, and computer science. In number theory, it is used to find solutions to Diophantine equations and to study residue classes. In cryptography, it is used to efficiently decrypt messages encrypted using multiple keys. In computer science, it is used in algorithms for efficient parallel processing and in error-correcting codes.

How is the Chinese Remainder Theorem related to the Chinese Remainder Theorem for polynomials?

The Chinese Remainder Theorem for polynomials is a generalization of the Chinese Remainder Theorem for integers. It deals with finding solutions to systems of polynomial congruences, where the divisors are pairwise relatively prime. The theorem states that if there are several polynomial congruences with different moduli, it is possible to combine them to form a single polynomial congruence with a larger modulus, and still preserve the solutions. This has important applications in algebraic geometry and coding theory.

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