This study presents an integer-only algorithm to exactly recover an image from its discrete projected views that can be computed with the same computational complexity as the fast Fourier transform (FFT). Most discrete transforms for image reconstruction rely on the FFT, via the Fourier slice theorem (FST), in order to compute reconstructions with lowcomputational complexity. Consequently, complex arithmetic and floating point representations are needed, the latter of which is susceptible to round-off errors. This study shows that the slice theorem is valid within integer fields, via modulo arithmetic, using a circulant theory of the Radon transform (RT). The resulting number-theoretic RT (NRT) provides a representation of images as discrete projections that is always exact and real-valued. The NRT is ideally suited as part of a discrete tomographic algorithm, an encryption scheme or for when numerical overflow is likely, such as when computing a large number of convolutions on the projections. The low-computational complexity of the NRT algorithm also provides an efficient method to generate discrete projected views of image data.