Suppose the force of gravity between two objects 1 and 2 of masses \(m_{1}\) and \(m_{2}\) were proportional to the sum \(m_{1}+m_{2}\) rather than to the product \(m_{1} m_{2}\).

(a) Would this dependence be consistent with the following two requirements? 

(i) The force exerted by 1 on 2 is equal in magnitude to the force exerted by 2 on 1. 

(ii) The force of gravity exerted by Earth on two identical objects is equal to the sum of the forces exerted on the individual objects. 

(b) Can you think of any other combination of \(m_{1}\) and \(m_{2}\) that satisfies requirements (i) and (ii)?

(c) Suppose you are given a lump of matter of mass M. How should you divide this lump into two parts to maximize the force of gravity between the two parts?