suppose a=3
b=5
ab=5*3=15
a+b=8
so a+b is not a relatively prime.
one could suppose that a counter-example existed. if so then since d > 1 divides ab and a+b, there is some prime p that divides d. but p, being prime, must divide a or b.
suppose it were a, that is a = kp. since p also divides a+b (since d does), a+b = rp. hence b = a+b - a = rp - kp = (r-k)p, so p divides b.
or, suppose it were b, so that b = mp. then a = a+b - b = rp - mp = (r-m)p, and so a is likewise divisible by p.
either way, assuming that gcd(ab,a+b) > 1 leads to a prime p dividing a and b. but gcd(a,b) = 1, so how can this be possible?