What is Prime Factorization?

  • Prime factorization is the process of expressing a number as a product of prime numbers.
  • Every composite number can be broken down uniquely into prime factors (except for the order).
  • This is known as the Fundamental Theorem of Arithmetic.

Importance in Exams

  • Used in finding HCF and LCM of numbers.
  • Helpful for solving divisibility, simplification, and factorization questions.
  • Prime factorization questions appear frequently in SSC, RRB, and Banking exams.

Basic Method: Division by Smallest Prime

  • Start dividing the number by the smallest prime number (2).
  • Continue dividing until it is no longer divisible by that prime.
  • Move to the next prime number (3, 5, 7, etc.).
  • Repeat until you reach 1.
  • Example: For 60:
  • 60 ÷ 2 = 30
  • 30 ÷ 2 = 15
  • 15 ÷ 3 = 5
  • 5 is prime.
  • Prime factors: 2 × 2 × 3 × 5.

Prime Factor Tree Method

  • Draw a factor tree by splitting the number into any two factors.
  • Continue splitting until all factors are prime numbers.
  • Example for 48:
  • 48 → 6 × 8
  • 6 → 2 × 3
  • 8 → 2 × 4
  • 4 → 2 × 2
  • Prime factors: 2 × 2 × 2 × 2 × 3.

Ladder (Division Ladder) Method

  • Write the number and divide successively by primes, arranging divisions in a step-like ladder.
  • Record each prime divisor on the side.
  • The list of side divisors gives the prime factors.
  • Example for 90:
  • Divide by 2: 90 ÷ 2 = 45
  • Divide by 3: 45 ÷ 3 = 15
  • Divide by 3: 15 ÷ 3 = 5
  • 5 is prime.
  • Prime factors: 2 × 3 × 3 × 5.

Properties of Prime Factorization

  • The product of prime factors equals the original number.
  • Prime factorization is unique (order may vary).
  • The exponents of primes can be used to find the number of divisors.

Using Exponents in Factorization

  • Instead of repeating primes, express them as powers.
  • Example: 2 × 2 × 2 × 3 = 2³ × 3.
  • This form is compact and often preferred in exams.

Shortcuts for Even Numbers

  • Keep dividing by 2 until the result is odd.
  • Then proceed with higher primes (3, 5, 7, etc.).

Special Cases

  • If the number itself is prime, its only factorization is 1 × itself.
  • Example: 13 → Prime factors: 13.

Prime Factorization for Large Numbers

  • Divide successively by primes up to the square root of the number.
  • If no divisor is found, the number is prime.

Application in HCF and LCM

  • HCF: Take the lowest power of common prime factors.
  • LCM: Take the highest power of all primes involved.
  • Example: 12 (2²×3) and 18 (2×3²):
  • HCF = 2×3=6.
  • LCM = 2²×3²=36.

Prime Factorization Table Method

  • Arrange divisions in a table format.
  • Left column: prime divisor.
  • Right column: quotient.
  • Continue until quotient is 1.

Prime Factorization Tips for Exams

  • Always check divisibility by 2, 3, 5, 7, 11 first.
  • Memorize primes up to 50.
  • Use exponents for clarity.
  • Recheck your factors to avoid mistakes.

Common Mistakes to Avoid

  • Forgetting to divide by the smallest primes first.
  • Stopping before all factors are prime.
  • Mixing up exponents and counts.

Miscellaneous Facts

  • Prime factorization is a building block for many number system topics.
  • Helps in simplifying fractions and algebraic expressions.
  • Important in cryptography and coding theory.

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