Classical multiplicative number theory with Euclid’s algorithm and continued fractions is presented anew in matrix formulation, which shows immediately, for instance, that there exist group structures over the integers. Very basics of modern sieve methods and prime number theory are also described so that readers can foresee well what will be developed in the analysis oriented final chapter. Continued fractions are presented as a device still fundamental in practical approaches to number theory, despite they are ignored in most modern treatises, which are often written with solely theoretical views. This chapter also describes a great historical tradition or cultural interactions encircling Euclid’s Elements, and how deeply we owe to the efforts of people in past ages.