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2. Let me N and a € Z. (a) If ged(a,m) = 1, then Bézout's lemma gives the existence of integers x and y such that ax + my = 1. Prove that x +mZ is the multiplicative inverse of a + mZ.

2. Let me N and a € Z. (a) If ged(a,m) = 1, then Bézout's lemma gives the existence of integers x and y such that ax + my = 1. Prove that x +mZ is the multiplicative inverse of a + mZ.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter1: Fundamental Concepts Of Algebra
Section1.1: Real Numbers
Problem 38E
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Please help with problem 2a, please provide as much detail as possible. 

2. Let meN and a € Z. (a) If ged(a,m) = 1, then Bézout's lemma gives the existence of integers x and y such that ax + my = 1. Prove that a+mZ is the multiplicative inverse of a +mZ. (b) Determine the least nonnegative integer representative for (11+163Z)-¹ by expressing 1 as a linear combination of 11 and 163 (using the extended Euclidean algorithm).
Transcribed Image Text:2. Let meN and a € Z. (a) If ged(a,m) = 1, then Bézout's lemma gives the existence of integers x and y such that ax + my = 1. Prove that a+mZ is the multiplicative inverse of a +mZ. (b) Determine the least nonnegative integer representative for (11+163Z)-¹ by expressing 1 as a linear combination of 11 and 163 (using the extended Euclidean algorithm).
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