Question

Let a, b be nonzero integers. Show that gcd(a,b)=1 if and only if gcd(a+b,ab)=1.

Answer 1

If gcd(a+b,ab)=p, p>1. Then if p |a from here p | a+b and p | b,
so gcd(a,b)=p but gcd(a,b)=1.
contradiction
Similarly to the other side.