Mathematicians have been fascinated by the concepts of elementary number theory, such as integers, integer division, prime and relatively prime numbers, sinc the times of Ancient Greece. Problems based on thes econcepts are easy to pose, but their solutions are often very complicated. Some of these problems, e.g. the twin prime hypothesis,though very old, still represent a great challenge for mathematicians. Elementary number theory might seem full of mathematical recreations which have been nothing more than a source of entertainment for centuries. Although we do not consider this purpose pointless, we shall try to convince the reader that many applications of everyday use are actually based on number theory principles. In fact, elementary number theory is a basic material for the construction of cryptographic systems used in e-mail communication or for credit card password authentication.