Chapter 1 : Divisibility and Factorization

The first chapter of the book introduces you to the basics of number theory, getting you comfortable with playing around with integers, introducing you to primes, and asking some basic questions about divisors and multiples. We'll cover roughly one section per day. The notes for each course will be posted as they become available.

- Lecture 0: An Introduction to Numbers
- We started today by getting to know the policies and expectations in the course. All of this is available already on the syllabus, but if you have any questions don't be shy about moc.liamg|ztluhcs.c.werdna#ydnA gniliame. We also spent some time introducing ourselves briefly; this will be continued as you post your own profiles for Homework 0. Afterwards, we started talking about the basics in number theory, starting with the axioms. We finished by introducing the notion of divisibility for the integers.
- Lecture 1: Introducing Divisibility
- Today we continued our discussion of divisibility and its basic properties. This culminated in the division algorithm. We also got a preview of the topic for next class: prime numbers.
- Lecture 2: Prime Numbers
- We went on a whirlwind introduction to prime numbers. After giving the basic definitions, we took a first step in showing that primes are the building blocks of the integers under multiplication. We then talked at length about their properties and how one can go about looking for prime numbers.
- Lecture 3: GCD and LCM
- Today we're switching gears and talking about divisors again, but this time we'll put a new spin on things by looking for common divisors of pairs of integers. This will lead us to the concept of greatest common divisor, a concept we'll explore in detail. We'll also talk about its cousin, the least common multiple of two integers.
- Lecture 4: The Euclidean Algorithm and the Fundamental Theorem of Arithmetic
- Today we began class by giving a computationally efficient way for computing GCDs known as the Euclidean Algorithm. In practice, this is how GCDs of large numbers are actually computed. Afterwards we discussed and proved the Fundamental Theorem of Arithmetic.
- Lecture 5: Primes in Arithmetic Progression; Modular Congruence
- We wrapped up our discussion on the Fundamental Theorem of Arithmetic, eventually hitting Dirichlet's Theorem on primes in an arithmetic progression. Afterwards we introduced the notion of modular congruence, the basic notion which drives modular arithmetic.

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