(i) if the gcd(a,b) = 1, then
(ii) if , then the gcd(a,b) = 1
Here you are trying to prove that if then there is no prime that divides both and . The neatest way to do this is by contradiction, suppose that and that there exists prime such that and and derive a contradiction.
Now you are trying to show that if for all primes that and then .(ii) if , then the gcd(a,b) = 1
Once again consider proof by contradiction ..
1) gcd(a,b)=1. Since all prime numbers > 1, no p exists that divides both a and b. No need to resort to contradiction imo
2) let no p divide both a and b and let gcd(a,b)=m. p divides m iff it divides both a and b. Since no p divides both a and b, then no p divides m. The only number that cannot be written as a product of primes is 1 so m =1