Although it is generally accepted that superconductivity is unconventional in the high-transition-temperature copper oxides, the relative importance of phenomena such as spin and charge (stripe) order, superconductivity fluctuations, proximity to a Mott insulator, a pseudogap phase and quantum criticality are still a matter of debate(1). In electron-doped copper oxides, the absence of an anomalous pseudogap phase in the underdoped region of the phase diagram(2) and weaker electron correlations(3,4) suggest that Mott physics and other unidentified competing orders are less relevant and that antiferromagnetic spin fluctuations are the dominant feature. Here we report a study of magnetotransport in thin films of the electron-doped copper oxide La2-xCexCuO4. We show that a scattering rate that is linearly dependent on temperature-a key feature of the anomalous normal state properties of the copper oxides-is correlated with the electron pairing. We also show that an envelope of such scattering surrounds the superconducting phase, surviving to zero temperature when superconductivity is suppressed by magnetic fields. Comparison with similar behaviour found in organic superconductors(5) strongly suggests that the linear dependence on temperature of the resistivity in the electron-doped copper oxides is caused by spin-fluctuation scattering.