How can i prove:
if gcd(a+b)=1 then gcd(a+b,ab)=1




I assume that what you want to prove is

if gcd(a,b)=1 then gcd(a+b,ab)=1

One method of proof is the following:

  • Assume that some integer greater than 1 divides a+b and ab, then there is a prime p that divides a+b and ab.
  • Since p is a prime and p divides a and b it divided either a or b.
    Assume p divides a.
  • Show that since p divided a and a+b it also divided b.
  • Thus gcd(a,b) ≥ p > 1