Prove Theorem 2.8 and Corollary J TICOCI 2.8.Theorem 2.8. For positive integers a and bgcd(a, b) lcm(a, b) = abProof. To begin, put d = gcd(a, b) and write a = dr. b = ds for integers r and 3. nala, then m = as = rb, the effect of which is to make m a (positive) commonmultiple 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 ratisiyingd = ax + by. In consequence,%3Dm =Ccdc (ах + by)x +y = vx + uy%3D%3DmababThis equation states that m | c, allowing us to conclude that m < c. Thus, in accordancewith Definition 2.4, m = lcm(a, b); that is,с.ababIcm(a, b) =d%3Dgcd(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, lcm(a, b) = ab if and only ifgcd(a, b) = 1.%3D

Name: Prove Theorem 2.8 and Corollary J TICOCI 2.8.Theorem 2.8. For positive
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: Prove Theorem 2.8 and Corollary J TICOCI 2.8.Theorem 2.8. For positive integers a and bgcd(a, b) lcm(a, b) = abProof. To begin, put d = gcd(a, b) and write a = dr. b = ds for integers r and 3. nala, then m = as = rb, the effect of which is to make m

Answered: Image /qna-images/answer/f6b2438a-9ccf-45fa-8edb-b9621e22df4e.jpg

Math

Advanced Math

Theorem 2.8. For positive integers a and b gcd(a, b) lcm(a, b) = ab Froof. To begin, put d = gcd(a, b) and write a = dr. b = ds for integersr and s. If m = ab/d, then m = as = multiple of a and b. Now let c be any positive integer that is a common multiple of a and D; say, for definiteness, c = au = bv. As we know, there exist integers x and y ratisfying d = ax + by. In consequence, rb, the effect of which is to make m a (positive) common cd c (ах + by) х + y = vx + uy %3D 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, %3D ab Icm(a, b) = d ab %3D %3D - 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, lcm(a, b) = ab if and only if gcd(a, b) = 1. %3D