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A new lattice Boltzmann model for multicomponent ideal gas mixtures is presented. The model development consists of two parts. First, a new kinetic model for Stefan- Maxwell diffusion amongst the species is proposed and realized as a lattice Boltzmann equation on the standard discrete velocity set. Second, a compressible lattice Boltzmann model for the momentum and energy of the mixture is established. Both parts are consistently coupled through mixture composition, momentum, pressure, energy and enthalpy whereby a passive scalar advection-diffusion coupling is obviated, unlike in previous approaches. The proposed model is realized on the standard three-dimensional lattices and is validated with a set of benchmarks highlighting various physical aspects of compressible mixtures. Stefan-Maxwell diffusion is tested against experiment and theory of uphill diffusion of argon and methane in a ternary mixture with hydrogen. The speed of sound is measured in various binary and ternary compositions. We further validate the Stefan-Maxwell diffusion coupling with hydrodynamics by simulating diffusion in opposed jets and the three-dimensional Kelvin-Helmholtz instability of shear layers in a two-component mixture. Apart from the multicomponent compressible mixture, the proposed lattice Boltzmann model also provides an extension of the lattice Boltzmann equation to the compressible flow regime on the standard three-dimensional lattice.