RRB NTPC & Group D Maths: Master LCM and HCF – Complete Guide with Shortcuts & Solved Questions

Introduction to LCM and HCF for RRB Exams

In the quantitative aptitude section of Indian Railway Recruitment Board (RRB) exams such as RRB NTPC, Group D, and Technician posts, Number Systems hold immense weightage. Within Number Systems, Least Common Multiple (LCM) and Highest Common Factor (HCF) are foundational concepts. Questions from this topic appear regularly in both CBT 1 and CBT 2 examinations, testing a candidate's ability to manipulate numbers efficiently under time constraints.

Understanding LCM and HCF goes beyond basic school arithmetic. It involves applying core properties to solve complex word problems relating to traffic lights ringing intervals, running tracks, fraction simplifications, and finding the greatest possible numbers that leave specific remainders. Mastering these concepts will give you a competitive edge in scoring full marks in this segment.

Topic Weightage and Importance

For RRB NTPC and Group D aspirants, Number Systems generally account for 2 to 4 questions per shift. Out of these, at least 1 or 2 direct or indirect questions are framed around LCM and HCF. While some questions are straightforward numerical calculations, others require logical application of concepts like consecutive numbers, co-prime numbers, and the relationship between the product of two numbers and their LCM and HCF. With proper shortcut methods, these questions can be solved within 30 to 45 seconds, saving crucial time for reasoning and general awareness sections.

Key Concepts and Formulas

Before diving into problem-solving, let us review the fundamental definitions and crucial mathematical formulas required for RRB exams:

• Highest Common Factor (HCF): Also known as the Greatest Common Divisor (GCD), the HCF of two or more numbers is the greatest number that divides each of them exactly without leaving a remainder.
• Least Common Multiple (LCM): The LCM of two or more numbers is the smallest positive number that is a multiple of each of the given numbers.
• Fundamental Formula for Two Numbers: For any two positive integers $a$ and $b$, the product of the numbers is equal to the product of their HCF and LCM:
  $$ \text{Product of } a \text{ and } b = \text{HCF}(a, b) \times \text{LCM}(a, b)$$
  $$a \times b = H \times L$$
• LCM and HCF of Fractions:
  $$ \text{HCF of Fractions} = \frac{ \text{HCF of Numerators}}{ \text{LCM of Denominators}}$$
  $$ \text{LCM of Fractions} = \frac{ \text{LCM of Numerators}}{ \text{HCF of Denominators}}$$
• Important Property: The HCF of two or more numbers is always a factor of their LCM. Also, the HCF of given numbers always divides their LCM completely.

Solved Examples (Step-by-Step)

Example 1: Basic Product Relationship

Question: The HCF of two numbers is 12 and their LCM is 72. If one of the numbers is 24, find the other number.

Solution:
Step 1: Write down the given values: $ \text{HCF} = 12$, $ \text{LCM} = 72$, First Number ($a$) = 24.
Step 2: Use the standard formula: $a \times b = \text{HCF} \times \text{LCM}$.
Step 3: Substitute the values: $24 \times b = 12 \times 72$.
Step 4: Solve for $b$:
$$b = \frac{12 \times 72}{24} = \frac{864}{24} = 36$$
Answer: The other number is 36.

Example 2: Finding HCF of Fractions

Question: Find the HCF of the fractions $\frac{2}{3}, \frac{4}{9}, \frac{5}{6}$.

Solution:
Step 1: Recall the formula for HCF of fractions: $ \text{HCF of fractions} = \frac{ \text{HCF of numerators}}{ \text{LCM of denominators}}$.
Step 2: Find the HCF of the numerators (2, 4, 5). Since there is no common prime factor among all three, their $ \text{HCF} = 1$.
Step 3: Find the LCM of the denominators (3, 9, 6).
$$ \text{Prime factorization: } 3 = 3^1, 9 = 3^2, 6 = 2^1 \times 3^1$$
$$ \text{LCM} = 2^1 \times 3^2 = 2 \times 9 = 18$$
Step 4: Combine them: $ \text{HCF} = \frac{1}{18}$.
Answer: $\frac{1}{18}$.

Example 3: Word Problem on Traffic Lights (LCM Application)

Question: Three electronic devices emit signals at intervals of 12 seconds, 18 seconds, and 27 seconds respectively. If they emit signals together right now, after what time will they emit signals together next?

Solution:
Step 1: Understand that the time when they emit signals together again will be a common multiple of the individual intervals. To find the *earliest* such time, we need to calculate the LCM of 12, 18, and 27.
Step 2: Perform prime factorization of each number:
$$12 = 2^2 \times 3^1$$
$$18 = 2^1 \times 3^2$$
$$27 = 3^3$$
Step 3: Take the highest power of each prime factor involved (2 and 3):
$$ \text{LCM} = 2^2 \times 3^3 = 4 \times 27 = 108 \text{ seconds}$$
Answer: They will emit signals together after 108 seconds (or 1 minute and 48 seconds).

Example 4: Greatest Number Leaving Remainders (HCF Application)

Question: Find the greatest number that will divide 43, 91, and 183 so as to leave the same remainder in each case.

Solution:
Step 1: When numbers $a, b, c$ leave the same remainder when divided by a common divisor, the differences between them ($(b-a)$, $(c-b)$, and $(c-a)$) are exactly divisible by that divisor.
Step 2: Calculate the differences:
$$91 - 43 = 48$$
$$183 - 91 = 92$$
$$183 - 43 = 140$$
Step 3: Find the HCF of these differences (48, 92, and 140).
$$48 = 2^4 \times 3$$
$$92 = 2^2 \times 23$$
$$140 = 2^2 \times 5 \times 7$$
Step 4: The highest common factor among 48, 92, and 140 is $2^2 = 4$.
Answer: The required greatest number is 4.

Common Mistakes to Avoid

• Confusing LCM and HCF in Word Problems: Students often mix up when to apply LCM vs HCF. Remember: if the question asks for the *smallest*, *least*, or *next simultaneous event*, use LCM. If it asks for the *greatest*, *largest possible division*, or *maximum capacity*, use HCF.
• Calculation Errors in Prime Factorization: Rushing through prime factorization leads to silly errors. Always cross-verify powers of prime numbers like 2, 3, and 5.
• Applying Product Formula to Three Numbers: The formula $ \text{Product} = \text{HCF} \times \text{LCM}$ is strictly valid for **two** numbers only. Do not apply it directly for three or more numbers.
• Ignoring Unit Conversions: In word problems, ensure all time or distance units are uniform before calculating LCM or HCF.

Practice Questions with Solutions

Q1: Find the LCM of 24, 36, and 40.
Q2: The ratio of two numbers is 3:4 and their HCF is 5. What is their LCM?
Q3: Find the greatest number of 4 digits which is divisible by 15, 25, 40, and 75.
Q4: Three runners start running around a circular track of length 1200 m at the same point with speeds of 2 m/s, 4 m/s, and 5 m/s in the same direction. After what time will they meet again at the starting point?
Q5: Find the HCF of 54, 288, and 360.

Solutions to Practice Questions:
Sol 1: Prime factorizations are $24 = 2^3 \times 3$, $36 = 2^2 \times 3^2$, $40 = 2^3 \times 5$. Taking highest powers: $2^3 \times 3^2 \times 5 = 8 \times 9 \times 5 = 360$.
Sol 2: Let numbers be $3x$ and $4x$. Since HCF is $x$, $x = 5$. Numbers are 15 and 20. $ \text{LCM} = \frac{15 \times 20}{ \text{HCF}} = \frac{300}{5} = 60$ (Alternatively, $ \text{HCF} \times \text{Product of ratios} = 5 \times 3 \times 4 = 60$).
Sol 3: First, find LCM of 15, 25, 40, and 75, which is 600. The greatest 4-digit number is 9999. Divide 9999 by 600, remainder is 399. Subtract remainder from 9999: $9999 - 399 = 9600$.
Sol 4: Time taken by each runner to complete one round: $T_1 = \frac{1200}{2} = 600$ s, $T_2 = \frac{1200}{4} = 300$ s, $T_3 = \frac{1200}{5} = 240$ s. Find LCM of 600, 300, and 240, which is 1200 seconds.
Sol 5: $54 = 2 \times 3^3$, $288 = 2^5 \times 3^2$, $360 = 2^3 \times 3^2 \times 5$. Common lowest powers give $ \text{HCF} = 2^1 \times 3^2 = 18$.

Frequently Asked Questions (FAQs)

Q1: Can the HCF of two numbers be greater than their LCM?

No, the HCF of two or more numbers can never be greater than their LCM. In fact, HCF is always less than or equal to the LCM.

Q2: Are LCM and HCF questions asked in RRB Group D CBT?

Yes, LCM and HCF form a core part of the mathematics syllabus for both RRB NTPC and RRB Group D exams. You can consistently expect 1 question in almost every shift.

Q3: What is the shortcut to find LCM quickly?

Instead of traditional division method, use the prime factorization method or pick the largest number among the given set and check its multiples until it is divisible by all other numbers.

Conclusion and Final Tips

Mastering LCM and HCF is a stepping stone toward conquering the Quantitative Aptitude section in RRB NTPC and Group D examinations. By understanding the underlying principles, recognizing the nature of word problems, and utilizing shortcut methods, you can solve these questions swiftly and accurately. Practice regularly, review past year papers, and time yourself to build exam-day confidence. Good luck with your preparation!