
Help with some Proofs
I came across these theorems, I was wondering if anyone could help with them!
1. Let a, b, n be elements of N with n >= 1. Prove that a and b are relatively prime iff a^n and b are relatively prime.
2. Suppose a, b, c are elements of N with at least one not zero. How would you define the greatest common divisor of a, b, c? Prove that it exists AND is unique.
I'm pretty sure for the 2nd one that gcd(a, b, c) = gcd(gcd(a,b), c). Any help actually PROVING these?

For the second one, I can prove the uniqueness easily. But proving the existence is more difficult to me.
I want to prove that there exists a gcd of just a,b because then I could extend that out easily to gcd(gcd(a,b), c).
So now I just need to know how can I prove that there exists a gcd of a and b?