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Résumé : We show how adding longer range hopping to the rotating Bose/Fermi Hubbard model leads to a massively degenerate manifold of single-particle ground states with wavefunctions identical to those making up the lowest Landau level of electrons in a magnetic field. We find that in the presence of local interactions, and at the appropriate filling factors, the exact many-body ground state is given by Laughlin's fractional quantum Hall wavefunctions. The hopping matrix elements in our model fall off as a Gaussian, and when the flux per plaquette is small compared to the fundamental flux quantum one only needs to include nearest and next nearest neighbor hoppings. We suggest how to engineer this model using atoms in optical lattices, and describe some of the observable consequences of the resulting fractional quantum Hall physics.