What is Prime Factorization?

  • Prime factorization is expressing a number as the product of its prime factors.
  • Example: 60 = 2 × 2 × 3 × 5.

Definition of HCF and LCM

  • HCF (Highest Common Factor) is the greatest number that divides all given numbers exactly.
  • LCM (Lowest Common Multiple) is the smallest number that is a multiple of all given numbers.

Steps to Find HCF Using Prime Factorization

  • Step 1: Find prime factors of each number.
  • Step 2: Identify the common prime factors.
  • Step 3: Multiply the lowest powers of these common factors.

Steps to Find LCM Using Prime Factorization

  • Step 1: Find prime factors of each number.
  • Step 2: List all prime factors involved.
  • Step 3: Multiply the highest powers of each prime factor.

Example for HCF and LCM

  • Numbers: 12 and 18
  • 12 = 2² × 3
  • 18 = 2 × 3²
  • HCF = 2¹ × 3¹ = 6
  • LCM = 2² × 3² = 36

Properties in Calculations

  • HCF × LCM = Product of the numbers
  • Example: 6 × 36 = 216 = 12 × 18
  • This property helps verify answers.

Advantages of Prime Factorization

  • Provides a systematic approach to HCF and LCM.
  • Reduces calculation errors.
  • Essential for complex and larger numbers.

Prime Factorization for Multiple Numbers

  • For three or more numbers, apply the same steps:
  • Find prime factors for each.
  • HCF: Use common primes with lowest exponents.
  • LCM: Use all primes with highest exponents.

Example with Three Numbers

  • Numbers: 8, 12, and 20
  • 8 = 2³
  • 12 = 2² × 3
  • 20 = 2² × 5
  • HCF = 2² = 4
  • LCM = 2³ × 3 × 5 = 120

Tips to Identify Prime Factors

  • Divide by smallest primes first: 2, 3, 5, 7, 11...
  • Continue until the quotient becomes 1.

Use of Exponents

  • Always express repeated primes as powers for clarity.
  • Example: 2 × 2 × 2 × 3 = 2³ × 3

Common Mistakes to Avoid

  • Not including all primes in LCM.
  • Using highest power in HCF instead of lowest.
  • Forgetting to verify with HCF × LCM = Product.

Benefits of Mastery

  • Increases accuracy and speed.
  • Helps in time-bound exams.
  • Strengthens overall number system concepts.

Miscellaneous Facts

  • Prime factorization is the foundation of arithmetic operations.
  • Used in coding theory, cryptography, and advanced math.
  • Important for LCM of fractions and decimals (convert to whole numbers first).

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