Chapter 4 Resources for Professor Baker's Math Class

Welcome to the resources page for Chapter 4! Here you'll find examples, notes, and other helpful materials to guide you through the key concepts of Number Theory and Cryptography. Let's dive in and master these fascinating topics!

Chapter 4 Overview

Chapter 4 covers fundamental concepts in number theory, including divisibility, modular arithmetic, and integer representations. It also introduces some basic cryptographic principles. Understanding these concepts is crucial for more advanced topics in mathematics and computer science.

Section 4.1: Divisibility and Modular Arithmetic

This section explores the core ideas of divisibility and modular arithmetic. Key concepts include:

  • Divisibility: Understanding when one integer divides another. For example, $a | b$ means "a divides b."
  • Modular Arithmetic: Performing arithmetic operations within a specific modulus. The notation $a \equiv b \pmod{m}$ means "a is congruent to b modulo m", i.e., $m | (a-b)$.
  • Division Algorithm: For any integers $a$ and $d$ (where $d > 0$), there exist unique integers $q$ and $r$ such that $a = dq + r$ and $0 \le r < d$. Here, $q$ is the quotient and $r$ is the remainder.

Example: Determine if 3 divides 478,125. The sum of the digits is $4 + 7 + 8 + 1 + 2 + 5 = 27$, which is divisible by 3. Therefore, $3 | 478125$.

Section 4.2: Integer Representations and Algorithms

This section covers different ways to represent integers, including binary, octal, and hexadecimal representations. It also delves into algorithms for converting between these representations.

  • Binary Representation: Representing integers using only 0s and 1s (base 2).
  • Octal Representation: Representing integers using digits from 0 to 7 (base 8).
  • Hexadecimal Representation: Representing integers using digits from 0 to 9 and letters A to F (base 16).
  • Conversion Algorithms: Efficient methods for converting between different integer representations.

Example: Find the decimal expansion of $(D5A3)_{16}$. This is $13 \cdot 16^3 + 5 \cdot 16^2 + 10 \cdot 16^1 + 3 \cdot 16^0 = 5328 + 1280 + 160 + 3 = 6771$.

Section 4.3: Primes and Greatest Common Divisors

This section focuses on prime numbers, greatest common divisors (GCD), and related concepts.

  • Prime Numbers: Integers greater than 1 that are only divisible by 1 and themselves.
  • Greatest Common Divisor (GCD): The largest integer that divides two or more integers. The Euclidean Algorithm is used to efficiently find the GCD.
  • Prime Factorization: Expressing an integer as a product of prime numbers.

Example: Find the prime factorization of 6600. $6600 = 66 \cdot 100 = 2 \cdot 3 \cdot 11 \cdot 2^2 \cdot 5^2 = 2^3 \cdot 3 \cdot 5^2 \cdot 11$.

PowerPoint for Chapter 4

Download the Chapter 4 PowerPoint presentation for a visual overview of the material. This presentation summarizes key concepts, definitions, and examples.

Answers for Even Problems in Chapter 4

Check your understanding by reviewing the answers to the even-numbered problems in Chapter 4. Use these solutions to identify areas where you need further practice.

Keep practicing and don't hesitate to ask questions in class or during office hours. Good luck with Chapter 4!