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.
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