Divisibility Rules

Overview of Number Types and Divisibility Rules

  • The lesson sets foundational concepts for understanding fractions and defines key number types and divisibility rules.

Key Concepts

  1. Definitions

    • Factor: A whole number that multiplies with another number to produce a given number. Every number has at least two factors.

    • Prime Number: A number that has exactly two factors: one and itself.

      • Example: The number 7.

      • Factors: 7 and 1.

      • Multiplication pairs: (1 × 7) and (7 × 1).

    • Composite Number: A number that has more than two factors, meaning it is divisible by more than one number.

      • Example: The number 6.

      • Factors: 1, 2, 3, and 6.

      • Multiplication pairs: (1 × 6), (2 × 3).

    • Exclusions: The numbers 0 and 1 are neither prime nor composite.

  2. Divisibility Rules: A method to determine if a number can be divided evenly into another number, which helps in identifying factors.

    • General Rule: If a number is divisible by another without leaving a remainder, then the second number is a factor of the first.

    • Composite Behavior: A number divisible by another is also divisible by that number's factors.

      • Example: A number divisible by 10 is also divisible by 5 and 2.

  3. Specific Divisibility Rules:

    • Divisibility by 2: A number is divisible by 2 if its last digit (ones digit) is even: 0, 2, 4, 6, or 8.

      • Example: The number 792 ends in 2 and is even, thus divisible by 2.

    • Divisibility by 3: Add all digits of the number. If the sum is divisible by 3, then the number is also divisible by 3.

      • Example Calculation:

      • For 741: 7 + 4 + 1 = 12; since 12 is divisible by 3, so is 741.

      • For 613: 6 + 1 + 3 = 10; since 10 is not divisible by 3, neither is 613.

Applications

  • Identifying Number Types: By assessing the number's factors, students can categorize numbers as prime or composite.

  • Learning Factoring Rules: Understanding prime factorization, greatest common factor, and least common multiple using the knowledge of factors and divisibility rules.

Summary of Lesson Objectives

  • By the end of the lesson, students should understand how to:

    • Identify and differentiate between prime and composite numbers.

    • Apply divisibility rules to determine factors.

    • Utilize factoring methods for calculations involving whole numbers: prime factorization, greatest common factor (GCF), and least common multiple (LCM).