We present measurements of the branching fractions of the decays B+-->eta'K+ and B0-->eta'K0. For B0-->eta(')K(0)(S) we also measure the time-dependent CP-violation parameters S eta'(K(0)(S)) and C eta'(K(0)(S)), and for B+-->eta'K+ the time-integrated charge asymmetry A(ch). The data sample corresponds to 88.9 x 10(6) BB pairs produced by e(+)e(-) annihilation at the Upsilon(4S). The results are B(B+-->eta'K+)=(76.9+/-3.5+/-4.4) x 10(-6), B(B0-->eta'K0)=(60.6+/-5.6+/-4.6) x 10(-6), S eta'(K(0)(S))=0.02+/-0.34+/-0.03, C eta'(K(0)(S))=0.10+/-0.22+/-0.04, and A(ch)=0.037+/-0.045+/-0.011.