GCD / LCM

Why are hash table sizes almost always prime? Why does the Euclidean algorithm for `gcd` run in `O(log(min(a, b)))` time despite using only subtraction or remainders? **Number Theory Fundamentals** is the branch of math that answers these questions and quietly powers a surprising amount of practical software, from hashing and cryptography to scheduling and competitive programming shortcuts. This lesson introduces the core building blocks. You will study **prime numbers** and how to test primality efficiently, the **divisibility** relation and quick rules for spotting divisors, **GCD** (greatest common divisor) and **LCM** (least common multiple) with the identity `lcm(a, b) = a * b / gcd(a, b)`, and the **Euclidean algorithm** for computing `gcd` in logarithmic time. You will also do **prime factorization** by trial division (`O(sqrt(n))`) and see how factorizations make divisor counting, simplification, and periodicity arguments straightforward. This lesson builds on **Discrete Mathematics Basics**, where you learned how to reason precisely about sets, relations, and logical claims. Number theory uses that same proof-style reasoning, but applied to the integers and their multiplicative structure. With primes, `gcd`, and factorization in your toolkit, you will be ready to take the premium step into **Modular Arithmetic**, where you will reuse all of these ideas to work with `mod`, modular inverses, and Fermat's Little Theorem.