Articles of euclidean algorithm

finding units of $ \mathbb{Z} {3}] $

In order took for units of $ \mathbb{Z} [ \sqrt[3]{3}] $ I am using a generalized Euclidean algorithm on three numbers. If $x \leq y \leq z$ then : $$ (x,y,z) \to \text{ sort } ( x, y ,z -y ) $$ The three numbers I will use are $1, \sqrt[3]{3},\sqrt[3]{9}$. Then if $z-y$ is […]

Using Extended Euclidean Algorithm to find multiplicative inverse

Having some trouble working my way back up the Extended Euclidean Algorithm. I’m trying to find the multiplicative inverse of $497^{-1} (mod 899)$. So I started working my way down first finding the gcd: \begin{align} 899&=497\cdot1 + 402\\ 497&=402\cdot1 + 95\\ 402&=95\cdot4 + 22\\ 95&=22\cdot4 + 7\\ 22&=7\cdot3 + 1 \end{align} Now I work my […]

Use Euclid's algorithm to find the multiplicative inverse $11$ modulo $59$

I was wondering if this answer would be correct the multiplicative of $11$ modulo $59$ would be $5$ hence $5\cdot11 \equiv 4 \pmod{59}$. Is this correct?

Generalisation of euclidean domains

Recently I wondered how dependent the definition of euclidean domains is of the co-domain of the norm-function. To be precise: Let’s define a semi-euclidean domain as a domain $R$ together with a norm $\delta : R \rightarrow \alpha$ for an ordinal $\alpha$ (or even the class of all ordinals) such that for $f,g \in R\ […]

Information about Problem. Let $a_1,\cdots,a_n\in\mathbb{Z}$ with $\gcd(a_1,\cdots,a_n)=1$. Then there exists a $n\times n$ matrix $A$ …

I would like to find some information about the following propositions, and unfortunately I haven’t been able to find any. Let $a_1,\dots,a_n\in\mathbb{Z}$ with $\gcd(a_1,\dots,a_n)=1$. Then there exists a matrix $A\in M_{n\times n}(\mathbb{Z})$ with first row $(a_1,\dots, a_n)$ such that $\det A=1$. Or in another case: Let $F$ be a field and $f_1,\dots,f_n\in\mathbb{F}[x_1,\dots, x_r]$ with $\gcd(f_1,\dots,f_n)=1$. […]

Finding $d=\gcd(a,b)$; finding integers $m$ and $n$: $d=ma+nb$

Let $a=8316$ and $b=10920$ a) Find $d=\gcd(a,b)$. greatest common divisor of $a$ and $b$ b) Find integers $m$ and $n$ such that $d=ma+nb$ this is what i’ve tried so far. correct me if I’m wrong 8316= 8016*4 + 300 10920= 10800*300 + 120 300= 120*2 + 60 60=30*2

Bezout's Theorem

I have seen the proof of Bezout’s theorem via the use of strong induction. \medskip The theorem states the following; Let $a$ and $b$ $\in \mathbb{Z}.$ Then there exists $m$, $n$ $\in \mathbb{Z}$ such that; $$gcd(a,b) = a(m) + b(n).$$ The proof starts with two base cases $r_0$ and $r_1$, where $r_i$ are the remainder […]

Find $g$ using euclids algorithm

Hi so I am doing this question trying to find the greatest common factor of $a = 44,238,054$ and $b = 4,753,791$ and this is what I got but I don’t think it is right. Was hoping someone could tell me what I’ve done wrong, thank you. 🙂 the rest of the question is: b) […]

Extended Euclidean Algorithm, what is our answer?

I am learning Euclidean Algorithm and the Extended Euclidean Algorithm. The problem I have is: Find the multiplicative inverse of 33 modulo n, for n = 1023, 1033, 1034, 1035. Now I learned that a multiplicative inverse only exists if the gcd of two numbers is 1. Thus, there doesn’t exist a multiplicative inverse for […]

Euclidean Algorithm help!

(A) Use the Euclidean Algorithm to find $\gcd (57, 139)$. (B) Use your work from part (a) to write your gcd as a linear combination of the two numbers. (C) Find the inverse of $57$ in $U(139)$. I know the gcd is $1$ and can do part (A) fine. I know I am supposed to […]