8. (a) Prove that if p and q are odd primes and q | a² - 1, then either q |a - 1 or else q=2kp + 1 for some integer k. [Hint: Because aP = 1 (mod q), the order of a modulo q is either 1 or p; in the latter case, p(q).] (b) Use part (a) to show that if p is an odd prime, then the prime divisors of 2P - 1 are of the form 2kp + 1. (c) Find the smallest prime divisors of the integers 2¹7 – 1 and 22⁹ (a) Vanif 0
8. (a) Prove that if p and q are odd primes and q | a² - 1, then either q |a - 1 or else q=2kp + 1 for some integer k. [Hint: Because aP = 1 (mod q), the order of a modulo q is either 1 or p; in the latter case, p(q).] (b) Use part (a) to show that if p is an odd prime, then the prime divisors of 2P - 1 are of the form 2kp + 1. (c) Find the smallest prime divisors of the integers 2¹7 – 1 and 22⁹ (a) Vanif 0
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.6: Inequalities
Problem 78E
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