Hi, I'm been doing some questions, and I am struggling with these 3, can someone please help? :) Problem 1: Let Zn denote the integers modulo n.(a) Write down tables for addition and multiplication in Zg.(b) Are there any zero divisors in Zg ?Problem 2: Using Euclid's algorithm, calculate the greatest common divisor of 144227 and58483 , and find integers x, y such that144227 x + 58483 y = gcd(144227, 58483)Problem 3: For each of the following linear congruences, either show that it has no solutions orfind all the mutually incongruent solutions.(а) 5а %3D 3 (mod 10);(b) 7х %3 48 (mod 53)%;

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Answered: Problem 1: Let Zn denote the integers…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.5: Congruence Of Integers

Problem 34E

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This question is related to unique factorisation in the Gaussian integers. In the Gaussian integers, find the gcd of 13 and 8 + i. The problem says "the gcd" (which is correct), but what are all the possibilities for the gcd?
q1. Find the integer a, if the modular representation of a is (2, 1, 5) with respect to the modulo 3, 5, 7?
For n∈N we define the set Z_n={1,2,…,n-1} and on this set we define the modular product as follows:for x,y,z∈Z_n ∶(x.y=z)⇔(x.y≡z mod n).In other words, we get the number (z) by calculating the product of the numbers x and y as a common product of two natural numbers, and from this product we then calculate the remainder after dividing by the number n. Examples for n = 5 and different values of x and y:in Z_5 ∶ 3.4=2 , 2.3=1 , 2,4=3 … We construct the graph G so that its vertices are elements of the set Z_101 and the two vertices corresponding to the elements x and y are joined by an edge just when the set Z_101 holds: x.y = 1 in the sense of the modular product defined above. Is graph G continuous? And continuous means like (continuing).Thanks
For n ∈ N we define the set Zn = {1, 2, . . , n - 1} and on this set we define the modular product as follows: For x, y, x ∈ Zn : (x.y = z) ⇔ (x.y ≡ z mod n). In other words, we get the number z by computing the product of x and y as the ordinary product of two natural numbers and then calculating the remainder of this product after dividing by n. Examples for n = 5 and different values of x and y: in Z5 : 3.4 = 2 , 2.3 = 1 , 2.4 = 3 . . . Problem: We construct a graph G such that its vertices are elements of the set Z101 and the two vertices corresponding to the elements x and y are connected by an edge if and only if on the set Z101 : x.y = 1 in the sense of the modular product defined above. a) Is the graph G ordinary? b) Is the graph G continuous? c) Is the graph of G a tree? d) What will be the sum of all the numbers in the adjacency matrix of graph G?
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Is the graph G ordinary? For n∈N we define the set Z_n={1,2,…,n-1} and on this set we define the modular product as follows:for x,y,z∈Z_n ∶(x.y=z)⇔(x.y≡z mod n).In other words, we get the number (z) by calculating the product of the numbers x and y as a common product of two natural numbers, and from this product we then calculate the remainder after dividing by the number n. Examples for n = 5 and different values of x and y:in Z_5 ∶ 3.4=2 , 2.3=1 , 2,4=3 … We construct the graph G so that its vertices are elements of the set Z_101 and the two vertices corresponding to the elements x and y are joined by an edge just when the set Z_101 holds: x.y = 1 in the sense of the modular product defined above.a) Is the graph G ordinary?
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Problem 1: Let Zn denote the integers modulo n. (a) Write down tables for addition and multiplication in Zg. (b) Are there any zero divisors in Zg ?