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

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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 ... user avatar Alp

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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 ... user avatar João Sá

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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? user avatar kmart875

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show that there are arbitrarily long sequences of consecutive positive integers which are not sums of two squares [duplicate]

Let $p_1, p_2, . . . , p_k$ be different prime numbers. By the Chinese Remainder theorem, show that for each $k ∈ N$ there exists an integer n such that $p_1 |n+1; p_2 |n+2; ...; p_k |n+k$ but $p^2_k$ ... user avatar Arch

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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 ... user avatar Katli

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1vote1answer60views

Does this system of congruences have a solution? [duplicate]

I have the following congruence equation system: $$ \left\{ \begin{array}{c} x \equiv 7 \pmod{7} \\ x \equiv 4 \pmod{12} \\ x \equiv 16 \pmod{21} \\\end{array} \right. $$ I understand that: $$x\equiv ... user avatar Boorger

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Prove that there exist primes $p_{i}$ such that $\prod_{i=1}^{k}p_{i} \mid \sum_{i=1}^{k}(p_{i})^{a_{i}}$

If $k\geq 3$ is a given positive integer, prove that there exist prime numbers $p_{1}<p_{2}<\cdots<p_{k}$ and positive integers $a_{1},a_{2},\cdots,a_{k}$, such that $$p_{1}p_{2}\cdots p_{k} ... user avatar math110

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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). user avatar Timothy Jason

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Chinese remainder theorem when modulus are not distinct, is my solution right? also - is this a good way to solve such kind of questions? [duplicate]

i learnt about the Chinese remainder theorem, and im trying to solve the following question: find the minimal solution x for (1)x = 11 mod 24 (2)x = 5 mod 18 (3)x = 5 mod 30 i know that in order to ... user avatar Guy

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Can you generalise the Chinese Remainder Theorem to noncommutative rings without identity?

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 $r_1, ..., r_n \in R$ ... user avatar Sasha

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Ring isomorphism between $(k[x,y]/(xy))/(x+y-a)$ and $k\times k$

Let $k$ be a field and let $a\in k$ be a non zero element. Consider the quotient ring $A = k[x,y]/(xy)$. If $f\in k[x,y]$ we denote by $\overline{f}$ its image in $A$. Now consider the quotient ring $... user avatar Albert

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Prove a number is pseudoprime with Sophie Germain

Let p be a Sophie Germain prime. Let q = 2p + 1. Let a = −q − 2p. Given that q ≡ 3 (mod p − 1), prove that pq is a pseudoprime to base a. I want to show pq is pseudoprime to base a but I am struggling ... user avatar ThomasL123

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Isomorphism between Rings using Chinese Remainder Theorem

Let $A_n(K)$ be the vector space of applications betwen $\mathbb{Z}_n$ and the field $K$, consider it as a ring with the convolution product $(f*g)(t) = \sum_{m \in \mathbb{Z}_n} f(t)g(t-m)$. If $K$ ... user avatar Ricardo

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Proof that the direct sum of two cyclic modules is cyclic

I've seen the proof that for gcd$(m,n)=1$ the product $\mathbb{Z}/{n \mathbb{Z}} \times \mathbb{Z}/{m \mathbb{Z}}$ is cyclic since it is isomorphic to $\mathbb{Z}/{nm \mathbb{Z}}$ by the Chinese ... user avatar NoodleNami

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Chinese remainder theorem without co-prime modulus, and unknown difference

How can I generalize the solution if the modulus is not co-prime. I am familiar with the following. $\newcommand{\lcm}{\mathrm{lcm}}$Suppose you have a system of two congruences $$\tag{two} \begin{... user avatar Moin Ahmed

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