The purpose of this article is to present a connection between school and university mathematics. We examine infinite periodic (non-pre-periodic) decimal expansions of rational numbers with mathematical rigor by using the university-level Number Theory and Group Theory. We will find there connections to various concepts and results concerning the multiplicative group modulo m. We illustrate and concretize our considerations, and propose some preliminary task ideas for all school levels. We also describe how the didactic mathematics programs are organized at the University of Tampere, Finland. We present the mathematical background needed in this subject in Appendix section at the end of the paper.