RRB NTPC & Group D Maths: Master HCF and LCM – Concepts, Shortcuts & Solved Questions

Introduction to HCF and LCM for RRB Exams

In the competitive landscape of Indian Railway Recruitment Board (RRB) exams, such as RRB NTPC, Group D, and Technician, the quantitative aptitude section holds significant weight. Among the fundamental topics, Highest Common Factor (HCF) and Least Common Multiple (LCM) are cornerstones. These concepts are not just standalone chapters but serve as essential tools for solving complex problems in fractions, time and work, pipes and cisterns, and even ratio and proportion.

Understanding HCF and LCM is about more than just memorizing definitions; it is about recognizing patterns in numbers and applying the right logic to solve real-world mathematical puzzles. Whether you are calculating the largest possible tile for a floor or determining when several bells will ring together, these concepts are your best allies.

Topic Weightage and Importance

For RRB NTPC (CBT-1 & CBT-2) and RRB Group D, the Mathematics section usually comprises 25 to 35 questions. Historically, 2 to 4 questions are directly based on HCF and LCM. However, the indirect application of these concepts is seen in at least 5-7 other questions across various topics.

Given the time constraints of RRB exams—where you often have less than a minute per question—mastering the shortcuts for HCF and LCM can save you precious seconds, allowing you to tackle more time-consuming sections like General Intelligence and Reasoning.

Key Concepts and Formulas

To master this topic, you must first clarify the definitions and the mathematical relationships between numbers.

1. Factors and Multiples

• Factor: A number that divides another number exactly without leaving a remainder. For example, factors of 12 are 1, 2, 3, 4, 6, and 12.
• Multiple: A number that is the product of a given number and an integer. For example, multiples of 5 are 5, 10, 15, 20, and so on.

2. Highest Common Factor (HCF)

Also known as Greatest Common Divisor (GCD), it is the largest number that divides two or more given numbers exactly.

Methods to find HCF:

• Prime Factorization Method: Express each number as a product of prime factors. The HCF is the product of the lowest powers of common prime factors.
• Division Method: Divide the larger number by the smaller one, then divide the previous divisor by the remainder until the remainder is zero. The last divisor is the HCF.

3. Least Common Multiple (LCM)

The smallest number which is exactly divisible by each of the given numbers.

Methods to find LCM:

• Prime Factorization Method: The LCM is the product of the highest powers of all prime factors involved in the numbers.
• Common Division Method (Shortcut): Arrange numbers in a row and divide by prime numbers starting from the smallest (2, 3, 5...) until no two numbers are divisible by the same prime.

4. Important Formulas

Solved Examples (Step-by-Step)

Example 1: Basic HCF and LCM

Question: Find the HCF and LCM of 24, 36, and 40.

Step 1: Prime Factorization
24 = 2³ × 3¹
36 = 2² × 3²
40 = 2³ × 5¹

Step 2: Calculate HCF
Common prime factor is 2. The lowest power of 2 common to all is 2². Since 3 and 5 are not common to all three, they are ignored.
HCF = 2² = 4.

Step 3: Calculate LCM
Take highest powers of all factors: 2³, 3², and 5¹.
LCM = 8 × 9 × 5 = 360.

Example 2: HCF and LCM of Fractions

Question: Find the LCM of 2/3, 8/9, and 16/81.

Solution:
LCM of fractions = (LCM of Numerators) / (HCF of Denominators)
Numerators: 2, 8, 16. LCM(2, 8, 16) = 16.
Denominators: 3, 9, 81. HCF(3, 9, 81) = 3.
Answer: 16/3.

Example 3: Application of Product Rule

Question: The HCF of two numbers is 11 and their LCM is 693. If one of the numbers is 77, find the other number.

Solution:
We know: First Number × Second Number = HCF × LCM
77 × Second Number = 11 × 693
Second Number = (11 × 693) / 77
Second Number = 693 / 7 = 99.
Answer: 99.

Example 4: Real-world Application (Bells Problem)

Question: Three bells toll at intervals of 9, 12, and 15 minutes respectively. If they start tolling together, after what time will they next toll together?

Solution:
In such problems, we need to find the LCM of the time intervals.
LCM of 9, 12, and 15:
9 = 3²
12 = 2² × 3¹
15 = 3¹ × 5¹
LCM = 2² × 3² × 5 = 4 × 9 × 5 = 180 minutes.
180 minutes = 3 hours.
Answer: They will toll together again after 3 hours.

Common Mistakes to Avoid

• Confusing HCF and LCM in Word Problems: Remember, if the question asks for the "Maximum," "Largest," or "Greatest" size that divides exactly, find the HCF. If it asks for the "Minimum," "Smallest," or "Least" time/number that is divisible, find the LCM.
• Forgetting to Simplify Fractions: Before finding the HCF/LCM of fractions, ensure they are in their simplest form (reduced to lowest terms).
• Decimal Errors: When dealing with decimals, students often forget to equalize the number of decimal places before calculation, leading to an incorrect power of 10 in the result.
• Calculation Speed: Many students use the long division method for simple numbers. Practice prime factorization mentally for numbers up to 100 to save time.

Practice Questions with Solutions

Q1. Find the greatest number which can divide 1354, 1866, and 2762 leaving the same remainder 10 in each case.
Q2. Two numbers are in the ratio 3 : 4. If their HCF is 4, find their LCM.
Q3. Find the smallest number which when divided by 20, 25, 35, and 40 leaves remainders 14, 19, 29, and 34 respectively.
Q4. The HCF of two numbers is 12 and their difference is 12. Which of the following could be the numbers? (a) 66, 78 (b) 70, 82 (c) 84, 96 (d) 94, 106
Q5. Find the HCF of 0.54, 1.8, and 7.2.

Solutions:

S1. Answer: 64. Subtract the remainder first: (1354-10), (1866-10), (2762-10) = 1344, 1856, 2752. Now find HCF of 1344, 1856, 2752. HCF is 64.

S2. Answer: 48. If ratio is 3:4 and HCF is 4, the numbers are (3×4) and (4×4) = 12 and 16. LCM of 12 and 16 is 48.

S3. Answer: 1394. Here, (20-14)=6, (25-19)=6, (35-29)=6, (40-34)=6. The common difference 'k' is 6. Required number = LCM(20, 25, 35, 40) - k. LCM is 1400. 1400 - 6 = 1394.

S4. Answer: (c) 84, 96. Both numbers must be divisible by the HCF (12). 84 and 96 are both multiples of 12 and their difference is 12.

S5. Answer: 0.18. Equalize decimal places: 0.54, 1.80, 7.20. Find HCF of 54, 180, 720. HCF(54, 180, 720) = 18. Put decimal back: 0.18.

Frequently Asked Questions (FAQs)

1. Can the LCM of two numbers be smaller than their HCF?

No. By definition, the HCF is a factor (divisor) and the LCM is a multiple. The LCM will always be greater than or equal to the HCF.

2. What is the HCF of two consecutive even numbers?

The HCF of any two consecutive even numbers (like 10 and 12, or 44 and 46) is always 2.

3. Why is HCF and LCM important for RRB Technician exams?

In Technician exams, many physics and trade-related problems (like gear teeth ratios or electrical frequency synchronization) rely on these mathematical principles for calculation.

Conclusion and Final Tips

Mastering HCF and LCM is a vital step toward clearing the RRB NTPC and Group D exams. It builds the numerical fluency required for more advanced topics. Remember the golden rule: Product of two numbers = HCF × LCM—this single formula solves nearly 40% of the questions from this chapter.

Keep practicing with different types of numbers, especially fractions and decimals. Consistency is key. You've got this! Stay focused on your goal, and the Indian Railways will soon be your workplace. Good luck!