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Search for Vector-Like Quarks using Trilepton and Same-Sign Dilepton Events in 20.3 fb⁻¹ of Proton-Proton Collisions at √S = 8 TeV with the ATLAS DetectorA search is presented for vector-like quarks using 20.3 fb⁻¹ of proton-proton collisions at a center-of-mass energy of √S = 8 TeV collected with the ATLAS detector in 2012 at the CERN Large Hadron Collider. Vector-like quarks are predicted to exist in many theories beyond the Standard Model of particle physics that attempt to resolve the hierarchy problem. Events are selected containing jets including at least one b-jet, sizable missing transverse momentum, large scalar sum of the jet and lepton transverse momenta, and either three leptons or two leptons with the same electric charge. Standard Model processes rarely produce this final state and production of vector-like quarks would lead to an enhanced rate of such events. The data are interpreted in the context of a variety of models including pair production of vector-like quarks T and B, which have the same electric charge as the corresponding Standard Model quarks and can appear in either SU(2) weak isospin singlets, doublets or triplets. In addition single and pair production of the vector-like quark T₅/₃ which appears in an SU(2) weak isospin doublet or triplet and has electric charge 5/3, is considered. A moderate excess of data above the SM background expectation is observed with a significance of less than two standard deviations. The data are used to set limits at 95% Confidence Level (CL) on the new heavy quark mass for the various VLQ models considered using the CL_S method. The vector-like quarks T and B in the singlet model are excluded at 95% CL below a mass of 0.59 TeV and 0.62 TeV respectively. The T₅/₃ is excluded at 95% CL below a mass of 0.74 TeV when only pair production is considered and below 0.75 TeV when both pair and single production are considered.