Explanation Proof: We will prove this by the principle of mathematical induction. For n = 2 , gcd ( a 1 , a 2 ) = s 1 a 1 + s 2 a 2 . Now, let the statement be true for k ≥ 2 . That is thegcd of k ≥ 2 integers is an integral linear combination of a 1 , a 2 , ........ a k . We will have to prove that the statement is true for n = k + 1 . Now, gcd ( a 1 a 2 ........ a k a k + 1 ) = gcd ( a 1 , gcd ( a 2 ........ a k a k + 1 ) ) By the induction hypothesis, gcd ( a 2 ........ a k a k + 1 ) = r 2 a 2 + ........ + r k + 1 a k + 1 for some integers r 2 , ........ r k + 1 . Now, gcd ( a 1 , gcd ( a 2 ........ a k a k + 1 ) ) = gcd ( a 1 , r 2 a 2 +

Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)

3rd Edition

ISBN: 9780134689555

Author: Edgar Goodaire, Michael Parmenter

Publisher: PEARSON

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Chapter 5.1, Problem 22E

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To prove: That gcd(a1a2........an) is an integral linear combination of a1,a2,........an for all n≥2; that is, prove that there exist integers s1,s2,........sn such that gcd(a1a2........an)=s1a1+s2a2+........+snan.

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Chapter 5.1, Problem 21E

Chapter 5.1, Problem 23E

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Chapter 5 Solutions

Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)

Ch. 5.1 - True/False Questions The statement i=1n(2i1)=n2...Ch. 5.1 - Prob. 2TFQ Ch. 5.1 - Prob. 3TFQ Ch. 5.1 - Prob. 4TFQ Ch. 5.1 - Prob. 5TFQ Ch. 5.1 - Prob. 6TFQ Ch. 5.1 - Prob. 7TFQ Ch. 5.1 - Prob. 8TFQ Ch. 5.1 - Prob. 9TFQ Ch. 5.1 - Prob. 10TFQ

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To prove: That gcd ( a 1 a 2 ........ a n ) is an integral linear combination of a 1 , a 2 , ........ a n for all n ≥ 2 ; that is, prove that there exist integers s 1 , s 2 , ........ s n such that gcd ( a 1 a 2 ........ a n ) = s 1 a 1 + s 2 a 2 + ........ + s n a n .

Solution Summary: The author explains how to prove mathrmgcd(a_1,

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To prove: That gcd(a1a2........an) is an integral linear combination of a1,a2,........an for all n≥2; that is, prove that there exist integers s1,s2,........sn such that gcd(a1a2........an)=s1a1+s2a2+........+snan.

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Chapter 5 Solutions