The problem of prime factorization is particularly important in fields such as cryptography, where it plays a crucial role, especially in the security of public key cryptosystems like RSA. There are numerous factorization algorithms that have been developed over time, each with varying levels of complexity. These algorithms have played a crucial role in fields like mathematics and cryptography, where prime factorization remains a key challenge. In this study, the continued fraction method one of the factorization methods, is examined. To highlight the importance of the continued fraction factorization method, a brief mention is made of RSA's vulnerability to attacks, such as Weiner's attack, which exploits small private keys. Our approach aims to enhance the efficiency of factorization by integrating this method with relevant theorems by giving concrete examples with detailed tables.