Overview

This topic explores the foundational properties of numbers and key principles from number theory. You'll learn about types of numbers, divisibility, prime factorization, greatest common divisors (GCD), least common multiples (LCM), and the basic laws that govern arithmetic operations. These concepts are crucial for understanding how numbers behave and relate to each other.

Key Concepts and Rules

• Types of Numbers: Natural, whole, integers, rational, irrational, real
• Properties of Operations: Commutative, associative, distributive properties
• Divisibility Rules: Patterns to determine if a number is divisible by another (e.g., divisible by 3 if digits add up to a multiple of 3)
• Prime Numbers: Whole numbers greater than 1 with only two factors (1 and itself)
• Prime Factorization: Expressing a number as a product of prime numbers
• Greatest Common Divisor (GCD): Largest number that divides two or more integers evenly
• Least Common Multiple (LCM): Smallest number that is a multiple of two or more integers
• Order of Operations: PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction)

Step-by-Step Example

Problem: Find the GCD and LCM of 18 and 24.

Step 1: Prime factorization:
18 = 2 × 3 × 3 = 2 × 3²
24 = 2 × 2 × 2 × 3 = 2³ × 3

Step 2: GCD: Multiply common prime factors with lowest exponents:
GCD = 2¹ × 3¹ = 6

Step 3: LCM: Multiply all prime factors with highest exponents:
LCM = 2³ × 3² = 72

Answer:
GCD = 6
LCM = 72

Quick Tip