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For Theorem 2.8 and it corollary give a simplified and short example in order to explain it very well as a presentation. Ps(short and well structured)

For Theorem 2.8 and it corollary give a simplified and short example in order to explain it very well as a presentation. Ps(short and well structured)

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 50E: Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation...
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For Theorem 2.8 and it corollary give a simplified and short example in order to explain it very well as a presentation. Ps(short and well structured)
As an example, the positive common multiples of the integers -12 and 30, 120, 180, ... ; hence, lcm(-12, 30) = 60. The following remark is clear from our discussion: given nonzero integers %3D and b, lcm(a, b) always exists and Icm(a, b) < |ab|. We lack a relationship between the ideas of greatest common divisor and common multiple. This gap is filled by Theorem 2.8. Theorem 2.8. For positive integers a and b gcd(a, b) lcm(a, b) = ab Proof. To begin, put d = gcd(a, b) and write a = dr,b= ds for integers r and s. m = ab/d, then m = as = rb, the effect of which is to make m a (positive) common multiple of a and b. Now let c be any positive integer that is a common multiple of a and b; say, for definiteness, c = au = bv. As we know, there exist integers x and y ratisfying d = ax + by. In consequence, %3D cd c(ax + by) ;)* * (E): y = vx +uy m ab ab This equation states that m | c, allowing us to conclude that m < c. Thus, in accordance with Definition 2.4, m = lcm(a, b); that is, ab lcm(a, b) = d ab gcd(a, b) which is what we started out to prove. Theorem 2.8 has a corollary that is worth a separate statement. Corollary. For any choice of positive integers a and b, Icm(a, b) = ab if and only if gcd(a, b) = 1. Perhaps the chief virtue of Theorem 2.8 is that it makes the calculation of the =ast common multiple of two integers dependent on the value of their greatest ommon divisor-which, in turn, can be calculated from the Euclidean Algorithm hen considering the positive integers 3054 and 12378, for instance, we found thet ed(3054, 12378) = 6; whence, 3054 12378 lcm(3054, 12378) = = 6300402 6. Before moving on to other matters, let us observe that the notion of ommon divisor can he extendod t
Transcribed Image Text:As an example, the positive common multiples of the integers -12 and 30, 120, 180, ... ; hence, lcm(-12, 30) = 60. The following remark is clear from our discussion: given nonzero integers %3D and b, lcm(a, b) always exists and Icm(a, b) < |ab|. We lack a relationship between the ideas of greatest common divisor and common multiple. This gap is filled by Theorem 2.8. Theorem 2.8. For positive integers a and b gcd(a, b) lcm(a, b) = ab Proof. To begin, put d = gcd(a, b) and write a = dr,b= ds for integers r and s. m = ab/d, then m = as = rb, the effect of which is to make m a (positive) common multiple of a and b. Now let c be any positive integer that is a common multiple of a and b; say, for definiteness, c = au = bv. As we know, there exist integers x and y ratisfying d = ax + by. In consequence, %3D cd c(ax + by) ;)* * (E): y = vx +uy m ab ab This equation states that m | c, allowing us to conclude that m < c. Thus, in accordance with Definition 2.4, m = lcm(a, b); that is, ab lcm(a, b) = d ab gcd(a, b) which is what we started out to prove. Theorem 2.8 has a corollary that is worth a separate statement. Corollary. For any choice of positive integers a and b, Icm(a, b) = ab if and only if gcd(a, b) = 1. Perhaps the chief virtue of Theorem 2.8 is that it makes the calculation of the =ast common multiple of two integers dependent on the value of their greatest ommon divisor-which, in turn, can be calculated from the Euclidean Algorithm hen considering the positive integers 3054 and 12378, for instance, we found thet ed(3054, 12378) = 6; whence, 3054 12378 lcm(3054, 12378) = = 6300402 6. Before moving on to other matters, let us observe that the notion of ommon divisor can he extendod t
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