  We have performed angle-resolved photoemission spectroscopy on the overdoped
Ba$_{0.3}$K$_{0.7}$Fe$_2$As$_2$ superconductor ($T_c$ = 22 K). We demonstrate
that the superconducting (SC) gap on each Fermi surface (FS) is nearly
isotropic whereas the gap value varies from 4.4 to 7.9 meV on different FSs. By
comparing with under- and optimally-doped Ba$_{1-x}$K$_x$Fe$_2$As$_2$, we find
that the gap value on each FS nearly scales with $T_c$ over a wide doping range
(0.25 $\textyen leq$ $x$ $\textyen leq$ 0.7). Although the FS volume and the SC
gap magnitude are strongly doping dependent, the multiple nodeless gaps can be
commonly fitted by a single gap function assuming pairing up to the
second-nearest-neighbor, suggesting the universality of the short-range pairing
states with the $s_{\yenpm}$-wave symmetry.
