1. GCD(a^n,b^n)

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

proof. Let $S = \{ n \in N : a^nv + b^nw = 1 , v,w \in Z \}$

1 is in S since ax + by = 1, let k be in S, now I have trouble trying to prove k+1 is in S.

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

proof. Let $S = \{ n \in N : a^nv + b^nw = 1 , v,w \in Z \}$

1 is in S since ax + by = 1, let k be in S, now I have trouble trying to prove k+1 is in S.
Are you allowed to use a prime factorization argument? It looks pretty easy using that.

-Dan

3. I'm not sure if I can do that, because this problem comes before the chapter involving prime.