Finding the Highest Common Factor (HCF)

Definition

• Highest Common Factor (HCF): The largest factor that a set of numbers share in common.
• Typically, HCF problems focus on pairs of numbers.

Method 1: Listing Factors

1. Example: Find the HCF of 20 and 28.

   • Factors of 20: 1, 2, 4, 5, 10, 20
   • Factors of 28: 1, 2, 4, 7, 14, 28
   • HCF: The largest common factor is 4.

2. Example: Find the HCF of 12 and 18.

   • Factors of 12: 1, 2, 3, 4, 6, 12
   • Factors of 18: 1, 2, 3, 6, 9, 18
   • HCF: The largest common factor is 6.

Method 2: Using Prime Factors

• Break down numbers into their prime factors.
• Identify common prime factors and multiply them.

Steps:

1. Prime Factors Example:

   • Prime factors of 12: 2, 2, 3
   • Prime factors of 18: 2, 3, 3
   • Common prime factors: 2 and 3
   • HCF: Multiply 2 × 3 = 6

2. Including Repeated Factors Example:

   • Change 18 to 20 (prime factors: 2, 2, 5)
   • Common prime factors: Two '2's
   • HCF: Multiply 2 × 2 = 4

Advanced Example:

• Find the HCF of 28 and 42 using Prime Factors:

  • Prime factors of 28: 2, 2, 7
  • Prime factors of 42: 2, 3, 7
  • Common prime factors: 2 and 7
  • HCF: Multiply 2 × 7 = 14

• Find the HCF of 132 and 420:

  • Common prime factors: 2, 2, 3
  • HCF: Multiply 2 × 2 × 3 = 12

Conclusion

• Understanding and utilizing both methods can simplify finding the HCF of numbers.
• Practice using both listing and prime factorization approaches to master identifying the HCF effectively.