4. An integer dЄZ\ {0} is said to be square-free if p² | d for any prime p. Letde Z\{0, 1} be square-free. Show that √√d & Q, where √d = i√√-d if d <0. Youmay assume that if a, b and r are integers, with a, r not both zero, r❘ab and thegreatest common divisor of a and r equal to 1, then r|b.Hint: The difficult case is d > 1. Write √d = a/b Є Q and aim for a contradictionby considering prime factors of a.

We shall be using basic divisibility property of prime numbers

4. An integer dЄ Z\ {0} is said to be square-free if p² | d for any prime p. Let de Z\{0, 1} be square-free. Show that √√d & Q, where √d=i√√d if d <0. You may assume that if a, b and r are integers, with a, r not both zero, rlab and the greatest common divisor of a and r equal to 1, then r|b. Hint: The difficult case is d > 1. Write √√d = a/b € Q and aim for a contradiction by considering prime factors of a.

4. An integer dЄ Z\ {0} is said to be square-free if p² | d for any prime p. Let de Z\{0, 1} be square-free. Show that √√d & Q, where √d=i√√d if d <0. You may assume that if a, b and r are integers, with a, r not both zero, rlab and the greatest common divisor of a and r equal to 1, then r|b. Hint: The difficult case is d > 1. Write √√d = a/b € Q and aim for a contradiction by considering prime factors of a.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.1: Real Numbers

Problem 35E

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