Mathematics

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How to Find HCF:

1. Prime Factorization:

   • Factor each number into its prime factors.
   • Identify the common factors and multiply them together.

2. Listing Factors:

   • List all factors of each number.
   • Find the largest factor that appears in both lists.

3. Euclidean Algorithm:

   • For two numbers $a$ and $b$, repeatedly replace $a$ with $b$ and $b$ with $a \mod b$ until $b$ becomes 0. The last non-zero value of $a$ is the HCF.

Example:

For 48 and 60:

• Prime Factorization:

  • 48: $2^4 \times 3$
  • 60: $2^2 \times 3 \times 5$
  • Common factors: $2^2 \times 3 = 12$ (HCF)

• Listing Factors:

  • Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
  • Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  • Common factors: 1, 2, 3, 4, 6, 12 (HCF is 12)

Applications:

• Simplifying fractions.
• Solving problems involving ratios.