Modular arithmetic studies arithmetic operations under remainders and divisibility. This topic introduces divisibility, congruences, and modular computations, which simplify calculations with large numbers and form the basis of many algorithms in computer science, including hashing, cryptography, and error detection.

This topic explores greatest common divisors (GCDs), the Euclidean algorithm, and their applications. It also introduces multiplicative inverses, linear congruences, and the Chinese Remainder Theorem, which are fundamental tools for solving number-theoretic problems efficiently in algorithms and cryptography.

This topic introduces cryptography as the study of secure communication over insecure channels. It covers secret-key cryptography, classical ciphers, key exchange problems, and public-key cryptography concepts, illustrating how number theory and modular arithmetic enable secure data transmission in modern systems.

This topic studies how to evaluate algorithm efficiency by analyzing running time and growth rates. It introduces asymptotic notation, such as Big-Theta, and applies these concepts to compare algorithms, emphasizing performance for large inputs rather than exact execution details.

Mathematical induction is a proof technique used to establish the truth of infinitely many statements. This topic introduces the principle of induction, its variants, and applications to proving formulas, inequalities, and properties of recursively defined structures common in mathematics and computer science.

Recursion defines objects and processes in terms of themselves. This topic introduces recursive definitions, recursive algorithms, and techniques for reasoning about their correctness and efficiency.