OK,A and B are vector fields having curl equal to zero.
We are to show div of (A+B)=0
Possibly the question is wrong.Except from some specific example it is not correct.Hence,cannot be proved.
Actually,since I wanted someone to say with me that this is wrong, posted this over here.But,unfortunately,the curl and div signs did not appear clearly.
div(A+B) = div A + div B = 0, so we must have div A = -div B or A = -B, but the information about their curl tells us nothing about their divergence, so this is not true. div rot A = 0, but this does not mean div A = 0.