In my paper, a divisibility rule named the Vezir function is described, which has been generalized for every integer and encompasses all previously established divisibility rules. Subsequently, the paper discusses other areas where this function is instrumental. These include simplifying ratios, conducting rationality tests, generating prime numbers, and more. Towards the end of the paper, a set of hypotheses is also presented. Among these, the most significant is the novel approach involving the application of the Vezir function for prime number generation.