Introduction to HCF and LCM for RRB Exams

In the quantitative aptitude section of RRB NTPC, Group D, and Technician exams, the topic of Highest Common Factor (HCF) and Least Common Multiple (LCM) holds significant importance. These fundamental concepts are essential for solving problems related to number systems, time and work, and circular motion. Understanding the relationship between these numbers allows you to solve complex arithmetic problems with speed and accuracy.

Topic Weightage and Importance

In most RRB examinations, you can expect 1 to 2 questions directly based on HCF and LCM. Furthermore, proficiency in these concepts is a prerequisite for tackling advanced questions in algebra and ratio-based arithmetic. Given the high competition, mastering these basics can be the difference between selection and rejection.

Key Concepts and Formulas

Highest Common Factor (HCF): The greatest number that divides each of the given numbers exactly. It is also known as the Greatest Common Divisor (GCD).

Least Common Multiple (LCM): The smallest number that is exactly divisible by each of the given numbers.

Important Formulas:

  • For any two numbers 'a' and 'b': HCF(a, b) × LCM(a, b) = a × b
  • HCF of fractions = (HCF of Numerators) / (LCM of Denominators)
  • LCM of fractions = (LCM of Numerators) / (HCF of Denominators)

Solved Examples (Step-by-Step)

Example 1: Find the HCF of 24, 36, and 48.

Step 1: Prime factorize each number: 24 = 2³ × 3; 36 = 2² × 3²; 48 = 2⁴ × 3.

Step 2: Pick the common prime factors with the lowest exponents: 2² × 3 = 12. So, HCF is 12.

Example 2: Find the LCM of 12, 15, and 20.

Step 1: Prime factorize: 12 = 2² × 3; 15 = 3 × 5; 20 = 2² × 5.

Step 2: Multiply the highest power of each prime factor present: 2² × 3 × 5 = 4 × 3 × 5 = 60. So, LCM is 60.

Common Mistakes to Avoid

  • Confusing HCF and LCM definitions during speed calculations.
  • Neglecting prime factorization; using manual division can lead to errors.
  • Forgetting to simplify fractions to their lowest terms before applying HCF/LCM formulas.
  • Overlooking the relationship: HCF × LCM = Product of two numbers.

Practice Questions with Solutions

Q1: Find the HCF of 16 and 40. Sol: 8. Q2: Find the LCM of 8, 12, 16. Sol: 48. Q3: The product of two numbers is 120 and their HCF is 2. Find LCM. Sol: 60. Q4: Find LCM of 2/3, 4/9, 5/6. Sol: 20/3. Q5: Find HCF of 0.6, 9.6, 0.12. Sol: 0.12.

Frequently Asked Questions (FAQs)

Q: Is HCF/LCM common in all RRB exams? Yes, it is a staple topic across all technical and non-technical railway exams. Q: Can I use the division method for HCF? Yes, the long division method is highly efficient for larger numbers. Q: How long should I spend on one question? Aim to solve these in under 45 seconds using shortcuts.

Conclusion and Final Tips

Mastering HCF and LCM requires regular practice of prime factorization and mental math. Focus on the relationship between the two and practice daily to improve your speed. Stay consistent and keep your calculations neat.