We discuss the theory of continued fractions. First, we deal with the general form of a continued fraction, then we show that every rational number p\q can be expressed as a finite continued fraction, whose coefficients aᵢ can be determined by applying the Euclidean algorithm to (p,q). However, the value of infinite continued fractions is irrational. After that we discuss the expansion of a real number as a continued fraction. We prove some properties of (finite, infinite, periodic) continued fractions, which we use in solving some practical problems. In particular, we focus on the linear Diophantine equation ax+by=c and the quadratic Diophantine equation x²=ax+1. We also give a detailed proof of Lagrange's theorem. This fact is then used as the key tool to determine the fundamental solution of the Pell's equations x²-Dy²=± 1. Then, we show how to obtain the other solutions of such an equation. Finally, we use the continued fraction of to factorize the large integer D....