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Let a, b E N, with a > 0 and b > 0. (a) Let p, q, r, s be the unique integers such that a = qb +r, 0

Let a, b E N, with a > 0 and b > 0. (a) Let p, q, r, s be the unique integers such that a = qb +r, 0

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.4: Prime Factors And Greatest Common Divisor
Problem 24E: Let (a,b)=1. Prove that (a,bn)=1 for all positive integers n.
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Let a, b E N, with a > 0 and b > 0. (a) Let p, q, r, s be the unique integers such that a = qb + r, 0 <r < b, b = pr + s, 0 <s < r. (In other words, r = a%band s = b%r.) Prove that s < . (b) Using part (a), prove that the Euclidean algorithm, applied to inputs a and b, halts in at most 2 log, (b) + 2 steps. One `step' is one application of the Division Algorithm (i.e. one computation of the % operator), and we assume that computing gcd(a, 1) or gcd(a, 0) takes 0 steps.
Transcribed Image Text:Let a, b E N, with a > 0 and b > 0. (a) Let p, q, r, s be the unique integers such that a = qb + r, 0 <r < b, b = pr + s, 0 <s < r. (In other words, r = a%band s = b%r.) Prove that s < . (b) Using part (a), prove that the Euclidean algorithm, applied to inputs a and b, halts in at most 2 log, (b) + 2 steps. One `step' is one application of the Division Algorithm (i.e. one computation of the % operator), and we assume that computing gcd(a, 1) or gcd(a, 0) takes 0 steps.
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