Relation between Primes: Proving (2m-1, 2mn-1/2m-1) = 1

[helgamauer]

Prove, that if
(m,n) = 1 // m and n are two different primes
then
(2^m -1, 2^mn -1/2^m -1) = 1

 

[Bill Simpson]

1: If m and n are two different primes then isn't it redundant to say their gcd is 1?

2: Do you really mean to say the gcd of an integer and a rational = 1? Or is 1/2^m a typographical error or do you mean something else by that?