We report measurements of the branching fractions and direct CP asymmetries (A_{CP}) for B->K pi, pi pi and KK decays (but not pi^0 pi^0) based on the final data sample of 772x10^6 B B-bar pairs collected at the Upsilon(4S) resonance with the Belle detector at the KEKB asymmetric-energy e^+ e^- collider. We set a 90% confidence-level upper limit for K^+- K^-+ at 2.0x10^{-7}; all other decays are observed with branching fractions ranging from 10^{-6} to 10^{-5}. In the $B^0/B^0-bar -> K^+-pi^-+ mode, we confirm Belles previously reported large A_{CP} with a value of -0.069 +- 0.014 +- 0.007 and a significance of 4.4 sigma. For all other flavor-specific modes, we find A_{CP} values consistent with zero, including A_{CP}(K^+- pi^0) = +0.043 +- 0.024 +- 0.007 with 1.8 sigma significance. The difference of CP asymmetry between B^+- -> K^+- pi^0 and B^0/B^0-bar -> K^+-pi^-+ is found to be Delta A_{K pi} equiv A_{CP} (K^+ pi^0) - A_{CP}(K^+ pi^-) = +0.112 +- 0.027 +- 0.007 with 4.0 sigma significance. We also calculate the ratios of partial widths for the B-> K pi decays. Using our results, we test the validity of the sum rule A_{CP}(K^+ pi^-) + A_{CP}(K^0pi^+) x Gamma(K^0pi^+)/Gamma(K^+ pi^-)} - A_{CP}(K^+ pi^0)x 2Gamma(K^+ pi^0)/Gamma(K^+ pi^-) - A_{CP}(K^0 pi^0)x 2Gamma(K^0 pi^0)/Gamma(K^+ pi^-) = 0 and obtain a sum of -0.270 +- 0.132 +- 0.060 with 1.9 sigma significance.
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