# Questions tagged [chinese-remainder-theorem]

For questions related to the Chinese Remainder Theorem and its applications.

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### Number of solutions of $y^2=x^3$ in $\mathbb{Z}_{57}$

The title explains the question. It was one of the 25 questions of a 3 hour olympiad, so I hope it is not too hard. The olympiad is for undergraduate students, so I also hope it doesn't use any "...
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### Integer GCD computation via Chinese Remainder Theorem

Is it possible to use Chinese Remainder Theorem to reconstruct the GCD of two integers from several GCDs of their modular representations (i.e. residues modulo pair-wise coprime integers)? For example:...
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### relationship between Chinese remainder theorem and roots of polynomials

According to https://eprint.iacr.org/2020/1481.pdf on page 9: What is $\mathbb{Z}_p[\eta]$? I mean, $\eta := [X \mod F_1(X)]$. Does this mean a polynomial evaluated at $\eta$? If so, which polynomial?...
1 vote
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### Proof of Kummer's Theorem in Janusz's Algebraic Number Fields

There is a theorem in Janusz's Algebraic Number Fields stated as follows: Kummer's Theorem: Let $R$ be a Dedekind ring with quotient field $K$ and $R'$ the integral closure of $R$ in a finite ...
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### Missing a detail about Chinese Remainder Theorem and $Z$ Ring isomorphisms.

I'm trying to prove that $\mathbb{Z}/m\mathbb{Z}\times\mathbb{Z}/n\mathbb{Z} \cong \mathbb{Z}/mn\mathbb{Z}$ holds only when $\gcd(m,n)=1$ or in simpler terms when $n,m$ are coprime integers. So far I ...
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### Remainder of a square

Let R represent the function that, given two inputs, a and b, returns the remainder of a when divided by b. E.g.: if $$a = bx + c$$, then $$R(a, x) = c$$ This remainder function has a lot of ...
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### Reverse a mod in an equation

Being quick and to the point, I have the following equation. (A * M)%F = B I want to solve for M. How do you move the modular F?
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### Find solution using infinite descent.

Can someone help with a task? Need to find a solution other than $(0,0,0)$ with infinite descent. $x,y,z\in\mathbb{Z}$. Any help would be appreciated. The equation is $x^2-3y^2=2z^2$. I tried to ...
1 vote
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### Chinese Remainder Theorem, discrete math problem [closed]

$5^{2003}$ $\equiv$ $3 \pmod 7$ $5^{2003}$ $\equiv$ $4\pmod{11}$ $5^{2003} \equiv 8 \pmod{13}$ Solve for $5^{2003}$ $\pmod{1001}$ (Using Chinese remainder theorem).
Ultimately, my question is: does the following theorem hold? Let $I_1, ..., I_n$ be ideals of some ring $R$, with $R = I_i + I_j$ for $1 \leq i < j \leq n$. Then for any $r_1, ..., r_n \in R$ ...