We present a 5.4$sigma$ detection of the pairwise kinematic Sunyaev-Zeldovich (kSZ) effect using Atacama Cosmology Telescope (ACT) and $it{Planck}$ CMB observations in combination with Luminous Red Galaxy samples from the Sloan Digital Sky Survey (SDSS) DR15 catalog. Results are obtained using three ACT CMB maps: co-added 150 GHz and 98 GHz maps, combining observations from 2008-2018 (ACT DR5), which overlap with SDSS DR15 over 3,700 sq. deg., and a component-separated map using night-time only observations from 2014-2015 (ACT DR4), overlapping with SDSS DR15 over 2,089 sq. deg. Comparisons of the results from these three maps provide consistency checks in relation to potential frequency-dependent foreground contamination. A total of 343,647 galaxies are used as tracers to identify and locate galaxy groups and clusters from which the kSZ signal is extracted using aperture photometry. We consider the impact of various aperture photometry assumptions and covariance estimation methods on the signal extraction. Theoretical predictions of the pairwise velocities are used to obtain best-fit, mass-averaged, optical depth estimates for each of five luminosity-selected tracer samples. A comparison of the kSZ-derived optical depth measurements obtained here to those derived from the thermal SZ effect for the same sample is presented in a companion paper.
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