Adv Alegbra HW Solutions 26

Adv Alegbra HW Solutions 26 - 26 inductive step, use...

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Unformatted text preview: 26 inductive step, use antanairesis and the defining recurrence, ( Fn +2 , Fn +1 ) = ( Fn +1 − Fn , Fn +1 ) = ( Fn , Fn +1 ) = 1. Here is a proof that is a variation of the same idea. Let n ≥ 1 be the smallest integer for which Fn +1 and Fn have gcd d > 1. We note that n > 1 because ( F2 , F1 ) = (1, 1) = 1, and so n − 1 ≥ 1. But if d is a common divisor of Fn +1 and Fn , then d divides Fn −1 = Fn +1 − Fn , so that ( Fn , Fn −1 ) = 1. This contradicts n being the smallest index for which ( Fn +1 , Fn ) = 1. 1.65 (i) Show that if d is the greatest common divisor of a1 , a2 , . . . , an , then d = ti ai , where ti is in Z for all i with 1 ≤ i ≤ n . Solution. The set I of all linear combinations ti ai of a1 , a2 , . . . , an , where ti is in Z for 1 ≤ i ≤ n , satisfies the conditions of Corollary 1.37. If d is the smallest positive element in I , then the proof of Theorem 1.35 can be modified to show that d is the gcd. (ii) Prove that if c is a common divisor of a1 , a2 , . . . , an , then c | d . Solution. The proof of Corollary 1.40 generalizes easily. 1.66 (i) Show that (a , b, c), the gcd of a , b, c, is equal to (a , (b, c)). Solution. It suffices to prove that any common divisor of a , b, c is a common divisor of a and (b, c), and conversely. But each of these statements is easy to prove. (ii) Compute (120, 168, 328). Solution. (120, 168, 328) = (120, (328, 168)) = (120, 8) = 8 1.67 (i) Consider a complex number z = q + i p, where q > p are positive integers. Prove that (q 2 − p2 , 2q p , q 2 + p2 ) is a Pythagorean triple by showing that |z 2 | = |z |2 . Solution. If z = q + i p, then |z 2 | = |z |2 , by part (i). Now z 2 = (q 2 − p2 ) + i 2q p , so that |z 2 | = (q 2 − p2 )2 + (2q p )2 . On the other hand, |z |2 = (q 2 + p2 )2 . Thus, if we define a = q 2 − p2 , b = 2q p , and c = q 2 + p 2 , then a 2 + b2 = c2 and (a , b, c) is a Pythagorean triple. (ii) Show that the Pythagorean triple (9, 12, 15) (which is not primitive) is not of the type given in part (i). ...
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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