Positive integers, their divisors and properties are widely used not only in Mathematics but also, in Informatics and Programming respectively. The research project contains a wide range of properties of the functions of the number, sum, and product of the divisors of a positive integer. The basic data on these functions, broadened and supplemented with the personally derived propositions, allow one to obtain new theoretical results, as well as to conduct the calculations more rationally. This was illustrated in the derivation of the new results and the solutions to the self-created problems. The resulting material can help solve practical and theoretical problems and be applied as a database for further research.