Jana,
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
Penny
