8 gcd(49, 210, 350) = 7The reader is cautioned that it is possible for three integers to be relatively prime asa triple (in other words, gcd(a, b, c) = 1), yet not relatively prime in pairs; this isbrought out by the integers 6, 10, and 15.castPROBLEMS 2.41. Find gcd(143, 227), gcd(306, 657), and gcd(272, 1479).2. Use the Euclidean Algorithm to obtain integers x and y satisfying the following:(a) gcd(56, 72) = 56x +72y.(b) gcd(24, 138) = 24x + 138 y.(c) gcd(119, 272) = 119x + 272y.(d) gcd(1769, 2378) = 1769x + 2378y.3. Prove that if d is a common divisor of a and b, then d = gcd(a, b) if and only ifgcd(a/d, b/d) = 1.[Hint: Use Theorem 2.7.]4. Assuming that gcd(a, b) = 1, prove the foliowing:(a) gcd(a +b, a – b) = 1 or 2.[Hint: Let d = gcd(a + b, a - b) and show that d 2a, d 2b, and thus thatd< gcd(2a, 2b) = 2 gcd(a, b).](b) gcd(2a +b, a + 2b) = 1 or 3.(c) gcd(a + b, a² + b²) = 1 or 2.[Hint: a? + b? = (a + b)(a – b) + 2b².](d) gcd(a + b, a² – ab + b²) = 1 or 3.[Hint: a? - ab +b² = (a + b)² –- 3ab.]5. For n > 1, and positive integers a, b, show the following:(a) If gcd(a, b) = 1, then gcd(a" , b") = 1.[Hint: See Problem 20(a), Section 2.2.](b) The relation a" | b" implies that a | b.[Hint: Put d = gcd(a, b) and write a = rd,b = sd, where gcd(r, s) = 1. By part (a),gcd(r", s") = 1. Show that r = 1, whence a = d.]6. Prove that if gcd(a, b) = 1, then gcd(a +b, ab) = 1.7. For nonzero integers a and b, verify that the following conditions are equivalent:(a) a|b.(b) gcd(a, b) = |a |.(c) lcm(a, b) =|b|.8. Find lcm(143, 227), lcm(306, 657), and lcm(272, 1479).9. Prove that the greatest common divisor of two positive integers divides their least commonmultiple.10. Given nonzero integers a and b, establish the following facts concerning lcm(a, b):(a) gcd(a, b) = lcm(a, b) if and only if a = ±b.(b) If k > 0, then lcm(ka, kb) = k lcm(a, b).(c) If m is any common multiple of a and b, then lcm(a, b) | m.[Hint: Put t = ]cm(a, b) and use the Division Algorithm to write m = qt +r, where0

Name: 8 gcd(49, 210, 350) = 7The reader is cautioned that it is possible for
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Description: 8 gcd(49, 210, 350) = 7The reader is cautioned that it is possible for three integers to be relatively prime asa triple (in other words, gcd(a, b, c) = 1), yet not relatively prime in pairs; this isbrought out by the integers 6, 10, and 15.castPROBLE