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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.