Let a, m, and n be positive integers with a>1. Prove that am – 1 | an – 1 if and only if m | n. For the "if" direction, write n = md with a positive integer and use the factorization amd – 1 = (am – 1) x (am(d-1) + am(d-2) + … + am + 1).

Let a, m, and n be positive integers with a>1. Prove that am – 1 | an – 1 if and only if m | n. For the "if" direction, write n = md with a positive integer and use the factorization amd – 1 = (am – 1) x (am(d-1) + am(d-2) + … + am + 1).

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.4: Prime Factors And Greatest Common Divisor

Problem 29E

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Let a, m, and n be positive integers with a>1. Prove that a^m – 1 | aⁿ– 1 if and only if m | n.

For the "if" direction, write n = md with a positive integer and use the factorization a^md – 1 = (a^m – 1) x (a^m(d-1) + a^m(d-2)+ … + a^m + 1).

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