The periodicity properties of Gauss sums allow us to factor integer numbers. We show that the excitation probability amplitudes of appropriate quantum systems interacting with specific laser fields are determined by Gauss sums. The resulting probabilities are experimentally accessible by measuring the fluorescence from this level. In particular, we discuss a two-photon transition in a ladder system driven by a chirped laser pulse. In addition, we consider two realizations of laser driven one-photon transitions. For each quantum system we demonstrate the power of this factorization scheme using numerical examples.