The relationship between Gilbert damping and magneto-crystalline anisotropy is studied here using an all-optical method in a perpendicular Co/Ni multilayer system by varying the Ti-buffer thickness. As the Ti-buffer thickness increases, the magneto-crystalline anisotropy is enhanced. The time-resolved Kerr signal of each sample is well described by its own intrinsic Gilbert damping constant in a wide range of the external magnetic field. Interestingly, we find that Gilbert damping constants increase linearly from 0.021 to 0.036 when the magneto-crystalline anisotropy of the samples varies from 2.4 to 3.4 Merg/cm(3). Such a linear relationship implies that the spin-orbit interaction is the main source of the damping process through spin-lattice relaxation in our system. (C) 2013 AIP Publishing LLC.