Foundations of Number Theory for Graduate and Postgraduate Entrance Exams

Navigating the rigorous mathematical demands of graduate and postgraduate entrance exams requires a rock-solid grasp of theoretical concepts and the ability to apply them to challenging problems. This written course is designed to build your mathematical reasoning from the ground up, ensuring you can tackle complex questions with confidence. You will transition from basic arithmetic properties to sophisticated algebraic structures, developing the precise analytical skills required for advanced academic pursuits. * Understand fundamental divisibility, the Euclidean algorithm, and the unique factorization of integers * Master modular arithmetic, linear congruences, and the Chinese Remainder Theorem * Apply key number-theoretic functions, including Euler's totient function, Möbius inversion, and Fermat's Little Theorem * Explore primitive roots, quadratic residues, and the Law of Quadratic Reciprocity * Solve classical Diophantine equations using structured algebraic methods * Practice mathematical proofs and reasoning structures common in advanced examinations This course begins with foundational definitions and key terminology before progressing systematically through divisibility, congruences, and multiplicative functions. Each section combines clear theoretical explanations with step-by-step written examples and practice problems designed to mirror the style of competitive entrance exams. This course is designed for university students, aspiring researchers, and anyone preparing for advanced mathematics examinations, requiring only a basic background in high school algebra. Begin reading today to elevate your mathematical problem-solving skills.