help 40. Using the method followed in Example 17, express thegreatest common divisor of each of these pairs of inte-gers as a linear combination of these integers.b) 33, 44e) 101, 203h) 3457, 4669a) 9, 11d) 21, 55g) 2002, 2339c) 35, 78f) 124, 323i) 10001, 13422 EXAMPLE 17 Express ged(252, 198) = 18 as a linear combination of 252 and 198 by working backwardsthrough the steps of the Euclidean algorithm.Solution: To show that ged(252, 198) = 18, the Euclidean algorithm uses these divisions:252 = 198 · 1+ 54198 = 54 · 3 + 3654 = 36 - 1+ 1836 = 18 - 2 + 0.We summarize these steps in tabular form:252198154119854362543618336182Using the next-to-last division (the third division), we can express gcd(252, 198) = 18 as a linearcombination of 54 and 36. We find that18 = 54 – 1. 36.The second division tells us that36 = 198 – 3 · 54.Substituting this expression for 36 into the previous equation, we can express 18 as a linearcombination of 54 and 198. We have18 = 54 – 1- 36 = 54 – 1 · (198 – 3 - 54) = 4 - 54 – 1· 198.The first division tells us that54 = 252 – 1. 198.Substituting this expression for 54 into the previous equation, we can express 18 as a linearcombination of 252 and 198. We conclude that18 = 4 - (252 – 1. 198) – 1 - 198 = 4 - 252 – 5 - 198,completing the solution.The next example shows how to solve the same problem posed in the previous exampleusing the extended Euclidean algorithm.

Name: help 40. Using the method followed in Example 17, express thegreatest
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Answered: 40. Using the method followed in…

Math

Calculus

40. Using the method followed in Example 17, express the greatest common divisor of each of these pairs of inte gers as a linear combination of these integers. a) 9, 11 b) 33, 44 c) 35, 78

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section: Chapter Questions

Problem 59RE

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