Question

if gcd(a,b)=1 then gcd(a^n,b^n)=1

Answer 1

Suppose that there is a prime \(p\) where \(p\) divides \(a^n\) and \(b^n\)(note that if a number has a divisor then there is a prime number that divides it). So \(p|a^n \) and \(b^n\) implies \(p|a\) and \(p|b\) so \(\gcd(a,b)>1\) a contradiction.

Answer 2

so are you saying that the original statement is invalid?

Answer 3

no, I am supposing it is invalid, and then showing a contradiction.