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Elastic scattering angular distributions for 12C on 12C, 40Ca, 90Zr, and 208Pb were measured for projectile energies between 10 and 35 MeV/nucleon. Total reaction cross sections were extracted by an optical model analysis and compared to the prediction of the Glauber model. This comparison suggests that the nucleus-nucleus total reaction cross section at intermediate and high energies is governed by the nucleon-nucleon cross section.