Please explain all steps and write clearly. Please do not copy from elsewhere and post solution as some of those are incorrect. Let

a,m

and

n

be positive integers with

a>1

. Prove that

a^(m)-1|a^(n)-1

\ if and only if

m|n

. [Hint: For the "if" direction, write

n=md

with

d

\ a positive integer and use the factorization

a^(md)-1=(a^(m)-1)\\\\times

\

{:(a^(m(d-1))+a^(m(d-2))+cdots+a^(m)+1)]

Let a,m and n be positive integers with a>1. Prove that am1an1 if and only if mn. [Hint: For the "if" direction, write n=md with d a positive integer and use the factorization amd1=(am1) (am(d1)+am(d2)++am+1)]