Articles of euclidean algorithm

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 […]

$15x\equiv 20 \pmod{88}$ Euclid's algorithm

Use Euclid’s algorithm to find a multiplicative inverse of 15 mod 88, hence solve the linear congruence $15x\equiv 20 \pmod{88}$ So far I have: $88=5\cdot15+13$ $15=1\cdot13+2$ Backwards substitution gives $15v+18w=1$ $1=15-1\cdot13$ $=15-1(88-5\cdot15)$ $=15\cdot15-1\cdot88$ now very stuck what to do from here:(

Proving the number of iterations in the Euclidean algorithm

Let m and n be natural numbers. Suppose $ min(m, n) \leq 2^k $ for some natural number k. Show that the Euclidean algorithm needs at most $ 2k $ iterations to find the GCD of m and n. Basically I have no clue how to start this proof, I think I should be looking […]