Ranks of least squares solutions of the matrix equation AXB=C

For a complex matrix equation AXB=C, we solve the following two problems: (1) the maximal and minimal ranks of least square solution X to AXB=C, and (2) the maximal and minimal ranks of two real matrices X0 and X1 in least square solution to AXB=C. We also give a necessary and sufficient condition for matrix equations to have a common least square solution.