The brochure is devoted to the following questions: can an arbitrary integer always be represented as a product of primes, and if so, how many decomposition methods exist? The answer to these questions is given by the main theorem of arithmetic, the proof of which is given in the book. In addition, the question of the existence of other "arithmetic" is considered, where this theorem is incorrect. The reader will not need knowledge different from those outlined in the school's mathematics course (except perhaps the checkmate method. induction).