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In this paper, a family of scaled factorized Broyden-like methods for unconstrained nonlinear least squares problems with zero or nonzero residual is presented. This family is based on a full rank factorized form of a structured quasi-Newton update and a scaled approximation of the second order term using given information. The resulting algorithms yield a q-quadratic convergence for the zero residual case and a q-superlinear convergence for the nonzero residual case, and maintain at least locally, the positive definiteness of the approximate Hessian matrices.