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Understanding HCF and LCM with Examples

Apr 29, 2025

Key Concepts: Finding HCF and LCM

Introduction

  • HCF (Highest Common Factor): The largest factor that divides two or more numbers.
  • LCM (Lowest Common Multiple): The smallest multiple that is exactly divisible by two or more numbers.
  • Strategy: Use prime factorization and Venn diagrams to find HCF and LCM efficiently.

Example 1: Numbers 42 and 56

  • Prime Factorization
    • 42: 2 x 3 x 7
    • 56: 2 x 2 x 2 x 7
  • Venn Diagram
    • Common factors: 2 and 7
    • Unique factors for 42: 3
    • Unique factors for 56: Two additional 2s
  • HCF
    • Product of the middle section: 2 x 7 = 14
  • LCM
    • Product of all factors: 2 x 2 x 2 x 7 x 3 = 168

Example 2: Numbers 60 and 72

  • Prime Factorization
    • 60: 2 x 2 x 3 x 5
    • 72: 2 x 2 x 2 x 3 x 3
  • Venn Diagram
    • Common factors: 2 x 2 x 3
    • Unique factors for 60: 5
    • Unique factors for 72: An additional 2 and a 3
  • HCF
    • Product of the middle section: 2 x 2 x 3 = 12
  • LCM
    • Product of all factors: 72 x 5 = 360

Application Example: Flashing Lights

  • Situation: Light A flashes every 12 seconds, Light B every 15 seconds
  • Prime Factorization
    • 12: 2 x 2 x 3
    • 15: 3 x 5
  • Venn Diagram
    • Common factor: 3
  • LCM
    • 12 x 5 = 60 seconds for both lights to flash together again.

Example 3: Buses Departing

  • Situation: Bus A leaves every 10 minutes, Bus B every 14 minutes
  • Prime Factorization
    • 10: 2 x 5
    • 14: 2 x 7
  • Venn Diagram
    • Common factor: 2
  • LCM
    • 14 x 5 = 70 minutes for both buses to leave together again.

Example 4: Three Numbers 8, 12, 18

  • Prime Factorization
    • 8: 2 x 2 x 2
    • 12: 2 x 2 x 3
    • 18: 2 x 3 x 3
  • Venn Diagram
    • Common factor for all: 2
    • Additional for 8 & 12: Another 2
    • Additional for 12 & 18: A 3
  • HCF
    • Common factor: 2
  • LCM
    • 18 x 2 x 2 = 72

Practice Problems

  1. Find the HCF of 72 and 96

    • Prime Factorization
      • 72: 2 x 2 x 2 x 3 x 3
      • 96: 2 x 2 x 2 x 2 x 2 x 3
    • Common factors: 2 x 2 x 2 x 3 = 24

  2. Find the LCM of 28 and 35

    • Prime Factorization
      • 28: 2 x 2 x 7
      • 35: 5 x 7
    • LCM: 28 x 5 = 140

Conclusion

  • Using prime factorization and Venn diagrams simplifies finding HCF and LCM.
  • Helps solve practical problems like synchronized timings or shared schedules.

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