Highest common factor

key notes :

What is the Highest Common Factor?

Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that can exactly divide two or more numbers without leaving a remainder.

Key Concepts

  1. Common Factor:
    • Definition: A number that divides two or more numbers exactly.
    • Example: For 12 and 18, common factors are 1, 2, 3, 6.
  2. Highest Common Factor:
    • Definition: The largest number that is a common factor of the given numbers.
    • Example: For 12 and 18, the HCF is 6.

How to Find the HCF

There are different methods to find the HCF:

1. Listing Factors Method

  1. List All Factors:
    • Find all the factors of each number.
    • Example: For 12 and 18:
      • Factors of 12: 1, 2, 3, 4, 6, 12
      • Factors of 18: 1, 2, 3, 6, 9, 18
  2. Identify Common Factors:
    • Find the numbers that appear in both lists.
    • Common Factors: 1, 2, 3, 6
  3. Select the Greatest One:
    • The HCF is the largest number in the list of common factors.
    • HCF: 6

2. Prime Factorisation Method

  1. Prime Factorise Each Number:
    • Break each number into prime factors.
    • Example:
      • 12 = 2² × 3
      • 18 = 2 × 3²
  2. Identify Common Prime Factors:
    • Find the common prime factors.
    • Common Prime Factors: 2 and 3
  3. Multiply the Lowest Powers:
    • Multiply the common prime factors, using the smallest power for each prime.
    • HCF: 2¹ × 3¹ = 6

3. Euclidean Algorithm (for Advanced Students)

  1. Divide the Larger Number by the Smaller Number and Find the Remainder:
    • Example: For 48 and 18:
      • 48 ÷ 18 = 2 remainder 12
  2. Replace the Larger Number with the Smaller Number and the Smaller Number with the Remainder:
    • New numbers: 18 and 12.
  3. Repeat Until the Remainder is 0:
    • Continue the process until the remainder is 0.
    • The divisor at this step is the HCF.

Examples

  1. Find the HCF of 24 and 36:
    • Listing Factors:
      • Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
      • Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
    • Common Factors: 1, 2, 3, 4, 6, 12
      • HCF: 12
    • Prime Factorisation:
      • 24 = 2³ × 3
      • 36 = 2² × 3²
      • Common Prime Factors: 2² ×3¹ =12
  2. Find the HCF of 56 and 84:
    • Listing Factors:
      • Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
      • Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
    • Common Factors: 1, 2, 4, 7, 14, 28
      • HCF: 28
    • Prime Factorisation:
      • 56 = 2³ × 7
      • 84 = 2² × 3 × 7
      • Common Prime Factors: 2² ×7¹ =28

Why HCF is Useful

Simplifying Fractions:

  • Helps in reducing fractions to their simplest form.

Solving Problems:

  • Useful in problems related to grouping or arranging items.

Dividing Objects:

  • Helps in dividing objects into equal parts without leftovers.

Practice Problems

  1. Find the HCF of 30 and 45.
  2. Find the HCF of 56 and 98.
  3. Find the HCF of 15 and 25.

Visual Representation

 12: 1, 2, 3, 4, 6, 12
 18: 1, 2, 3, 6, 9, 18

Common Factors: 1, 2, 3, 6
Highest Common Factor: 6

Prime Factorisation Example

12 = 2² × 3

18 = 2 × 3²

Common Prime Factors: 2 and 3

HCF = 2¹ × 3¹ = 6

let’s practice!