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The origin of the diverse population of galaxy clusters remains an unexplained aspect of large-scale structure formation and cluster evolution. We present a novel method of using X-ray images to identify cool core (CC), weak cool core (WCC), and non cool core (NCC) clusters of galaxies, that are defined by their central cooling times. We employ a convolutional neural network, ResNet-18, which is commonly used for image analysis, to classify clusters. We produce mock Chandra X-ray observations for a sample of 318 massive clusters drawn from the IllustrisTNG simulations. The network is trained and tested with low resolution mock Chandra images covering a central 1 Mpc square for the clusters in our sample. Without any spectral information, the deep learning algorithm is able to identify CC, WCC, and NCC clusters, achieving balanced accuracies (BAcc) of 92%, 81%, and 83%, respectively. The performance is superior to classification by conventional methods using central gas densities, with an average BAcc = 81%, or surface brightness concentrations, giving BAcc = 73%. We use Class Activation Mapping to localize discriminative regions for the classification decision. From this analysis, we observe that the network has utilized regions from cluster centers out to r~300 kpc and r~500 kpc to identify CC and NCC clusters, respectively. It may have recognized features in the intracluster medium that are associated with AGN feedback and disruptive major mergers.