The aim of the present paper is to study the dynamics and fixed points of a class of functions induced by integers and then apply the fixed point theorem to obtain general tests of divisibility of numbers. In the second part, we generalize the theorems on divisibility to obtain conditions of divisibility of polynomials by (x4 + q) type polynomials. We not only obtain generalizations of the well-known remainder theorem and the factor theorem of algebra but also compute explicit value of the quotient. The results are computationally explicit and can be easily implemented on digital computers.