We perform extensive magnetic Feshbach spectroscopy of an ultracold mixture of fermionic 40K and bosonic 87Rb atoms. The magnetic-field locations of 14 interspecies resonances is used to construct a quantum collision model able to predict accurate collisional parameters for all K-Rb isotopic pairs. In particular we determine the interspecies s-wave singlet and triplet scattering lengths for the 40K-87Rb mixture as -111 +/- 5 Bohr and -215 +/- 10 Bohr respectively. We also predict accurate scattering lengths and position of Feshbach resonances for the other K-Rb isotopic pairs. We discuss the consequences of our results for current and future experiments with ultracold K-Rb mixtures.
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