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The weighted MAX k-CUT problem consists of finding a k-partition of a given weighted undirected graph G(V,E) such that the sum of the weights of the crossing edges is maximized. The problem is of particular interest as it has a multitude of practical applications. We present a formulation of the weighted MAX k-CUT suitable for running the quantum approximate optimization algorithm (QAOA) on noisy intermediate scale quantum (NISQ)-devices to get approximate solutions. The new formulation uses a binary encoding that requires only |V|log_2(k) qubits. The contributions of this paper are as follows: i) A novel decomposition of the phase separation operator based on the binary encoding into basis gates is provided for the MAX k-CUT problem for k >2. ii) Numerical simulations on a suite of test cases comparing different encodings are performed. iii) An analysis of the resources (number of qubits, CX gates) of the different encodings is presented. iv) Formulations and simulations are extended to the case of weighted graphs. For small k and with further improvements when k is not a power of two, our algorithm is a possible candidate to show quantum advantage on NISQ devices.