1.1 Introduction to Whole Numbers

Whole numbers form the foundation of our number system. They're the building blocks we use for counting, measuring, and everyday math. Understanding place value, multiples, and divisibility rules helps us work with these numbers more efficiently.

Prime factorization breaks numbers down to their basic components. This skill is crucial for finding common factors and multiples, which come in handy for simplifying fractions and solving real-world problems involving groups and schedules.

Understanding Whole Numbers

Place value in whole numbers

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• Digits in a whole number have a place value determined by their position from right to left
• Place values based on increasing powers of 10 (ones, tens, hundreds, thousands)
• Whole numbers can be written in various forms
  • Standard form (1,234)
  • Word form (one thousand two hundred thirty-four)
  • Expanded form (1000 + 200 + 30 + 4)
• Comparing and ordering whole numbers involves analyzing place values from left to right
  • Number with greater value in leftmost place is larger
  • If leftmost place values equal, compare next place value to right
• Real-world applications of whole numbers include money (dollars and cents), measurements (length, weight, time), population counts, and inventory quantities
• Number line helps visualize whole numbers and their relationships

Multiples and divisibility rules

• Multiples are products of a number and any integer (3, 6, 9, 12, 15 are multiples of 3)
• Divisibility rules determine if a number is divisible by another number
  • Divisible by 2 if last digit is 0, 2, 4, 6, or 8 (even numbers)
  • Divisible by 3 if sum of digits is divisible by 3
  • Divisible by 4 if last two digits form a number divisible by 4
  • Divisible by 5 if last digit is 0 or 5
  • Divisible by 6 if divisible by both 2 and 3
  • Divisible by 9 if sum of digits is divisible by 9
  • Divisible by 10 if last digit is 0

Types of Whole Numbers

• Natural numbers (counting numbers) are positive integers starting from 1
• Whole numbers include natural numbers and zero
• Odd numbers are integers not divisible by 2
• Even numbers are integers divisible by 2

Prime Factorization and Least Common Multiples

Prime factors and common multiples

• Prime numbers have exactly two factors: 1 and itself (2, 3, 5, 7, 11, 13, 17)
• Composite numbers have factors other than 1 and itself (4, 6, 8, 9, 10, 12, 14, 15)
• Prime factorization expresses a composite number as a product of its prime factors
  1. Divide number by smallest prime factor possible
  2. Repeat until quotient is prime
  3. Express using exponents for repeated factors ($36 = 2^2 \times 3^2$)
• Greatest common factor (GCF) is largest factor two or more numbers have in common
  • Find by listing prime factors of each number and multiplying common factors using lowest exponent
• Least common multiple (LCM) is smallest positive number that is a multiple of two or more numbers
  • Find by listing prime factors of each number and multiplying all factors using highest exponent
  • Formula: $LCM(a, b) = \frac{|a \times b|}{GCF(a, b)}$
• Rounding and estimation are useful techniques for working with approximate values of whole numbers