(i) if the gcd(a,b) = 1, then
Since p|p,
(ii) if , then the gcd(a,b) = 1
Correct?
Try including some words explaining what you are trying to achive (preferably at each step).
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
Correct?
Once again consider proof by contradiction ..
CB
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
Q.E.D