Questions on LCM and HCF

In competitive exams such as those conducted by SSC (Staff Selection Commission) and RRB (Railway Recruitment Board), mathematical proficiency is crucial for achieving high scores. Among the various mathematical concepts tested, the Highest Common Factor (HCF) and Least Common Multiple (LCM) are essential topics that frequently appear in these exams.Here are 10 multiple-choice questions on LCM and HCF with detailed answers

• Highest Common Factor (HCF): Also known as the Greatest Common Divisor (GCD), the HCF of two or more numbers is the largest integer that divides all of them without leaving a remainder. Mastery of HCF is vital for solving problems related to divisibility, simplifying fractions, and understanding the underlying structure of number systems.
• Least Common Multiple (LCM): The LCM of two or more integers is the smallest positive integer that is divisible by each of them. Proficiency in finding the LCM is important for solving problems involving synchronization, scheduling, and determining common multiples in practical scenarios.

This article presents a series of carefully crafted sample questions on HCF and LCM, designed specifically for SSC and RRB competitive exams. Each question is followed by a detailed explanation to ensure a thorough understanding of the concepts and techniques involved. Whether you are preparing for an upcoming exam or seeking to strengthen your mathematical skills, these sample questions offer a valuable opportunity to practice and refine your problem-solving abilities in the context of competitive assessments.

Questions

1. What is the HCF of 24 and 36?

• • (a) 6
  • (b) 12
  • (c) 18
  • (d) 24

(b) 12
Prime Factorization:
$24 = 2^3 \times 3$
$36 = 2^2 \times 3^2$
HCF: Lowest powers of common prime factors: $2^2 \times 3 = 12$

2.What is the LCM of 15 and 25?

• (a) 75
• (b) 100
• (c) 50
• (d) 25

(a) 75
Prime Factorization:
$15 = 3 \times 5$
$25 = 5^2$
LCM: Highest powers: $3 \times 5^2 = 75$

3.Find the HCF of 45 and 60.

• (a) 15
• (b) 10
• (c) 30
• (d) 5

(a) 15
Prime Factorization:
$45 = 3^2 \times 5$
$60 = 2^2 \times 3 \times 5$
HCF: Common factors: $3 \times 5 = 15$

4.Find the LCM of 8 and 12.

• (a) 24
• (b) 48
• (c) 16
• (d) 32

(a) 24
Prime Factorization:
$8 = 2^3$
$12 = 2^2 \times 3$
LCM: Highest powers: $2^3 \times 3 = 24$

5. What is the HCF of 56, 84, and 126?

• (a) 14
• (b) 28
• (c) 7
• (d) 21

(b) 14
Prime Factorization:
$56 = 2^3 \times 7$
$84 = 2^2 \times 3 \times 7$
$126 = 2 \times 3^2 \times 7$
HCF: Lowest powers of common factors: $2 \times 7 = 14$

6.Find the LCM of 40,36 and 126.

• (a) 2220
• (b) 2520
• (c) 2624
• (d) 2020 

(b) 2520.

To find the LCM (Least Common Multiple) of the numbers 40, 36, and 126, we use their prime factorizations:

• • $40 = 2^3 \times 5$
    
  • $36 = 2^2 \times 3^2$
    
  • $126 = 2 \times 3^2 \times 7$
    

  To calculate the LCM, we take the highest power of each prime factor present in the factorizations:

  • The highest power of 2 is $2^3$.
  • The highest power of 3 is $3^2$.
  • The highest power of 5 is $5.$
  • The highest power of 7 is $7$.

  So, the LCM is:

  $$LCM=2^3\times 3^2\times 5\times 7=8\times 9\times 5\times 7=2520$$

  Thus, the correct answer is (b) 2520.

7. Find the LCM of 40,36 and 126.

• (a) 1580
• (b) 1680
• (c) 1720
• (d) 1600

(b) 1680

To find the LCM (Least Common Multiple) of the numbers 112, 140, and 168, we first determine their prime factorizations:

• • $112 = 2^4 \times 7$
    
  • $140 = 2^2 \times 5 \times 7$
    
  • $168 = 2^3 \times 3 \times 7$
    

  To calculate the LCM, we take the highest power of each prime factor present in the factorizations:

  • The highest power of 2 is $2^4$.
  • The highest power of 3 is $3.$
  • The highest power of 5 is $5$.
  • The highest power of 7 is $7$.

  So, the LCM is:

  $$LCM=2^4\times 3\times 5\times 7=16\times 3\times 5\times 7=1680$$

  Thus, the correct answer is (b) 1680.

8. Find the LCM of 2.4, 0.36 and 7.2.

• (a) 7.2
• (b) 5.2
• (c) 1.2
• (d) 4.2

(a) 7.2.

Here’s a step-by-step breakdown of how to find the LCM of the decimal numbers 2.4, 0.36., and 7.2:

1. 1. Convert decimals to whole numbers:

      • $2.4$ becomes $240$ by multiplying by $100$.
      • $0.36$ becomes $36$ by multiplying by $100$.
      • $7.2$ becomes $720$ by multiplying by $100$.

   2. Find the LCM of these whole numbers:

      • $240 = 2^4 \times 3 \times 5$
        
      • $36 = 2^2 \times 3^2$
      • $720 = 2^4 \times 3^2 \times 5$
        

      The LCM of $240$, $36$, and $720$ is $720$.

   3. Convert back to the original scale:

      • Since we multiplied by $100$, we now divide the LCM $720$ by $100$to get back to the original decimal scale.

   Therefore, the LCM of $2.4$, $0.36$ and $7.2$is $7.2$
   

   The correct answer is (a) 7.2.

9. Find the HCF of 3/4, 5/6 and 6/7

• (a)1/84
• (b)1/42
• (c)1/21
• (d)5/42

(a) 1/84.

To find the HCF (Highest Common Factor) of the fractions $\frac{3}{4}$, $\frac{5}{6}$, and $\frac{6}{7}$, we use the following formula:

$$\text{HCF of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of denominators}}$$

Step 1: Find the HCF of the numerators (3, 5, 6)

• The HCF of 3, 5, and 6 is 1 because 1 is the only common factor of these numbers.

Step 2: Find the LCM of the denominators (4, 6, 7)

• Prime factorization:

  • $4 = 2^2$
    
  • $6 = 2 \times 3$
    
  • $7 = 7$
    

• The LCM of 4, 6, and 7 is:

$$LCM = 2^2 \times 3 \times 7 = 84$$

Step 3: Calculate the HCF of the fractions

$$\text{HCF}=\frac{1}{\mathrm{84<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>}}$$

Thus, the correct answer is (a) 1/84.

10. The LCM of two numbers is 64699, their GCM or (HCF) is 97 and one of the numbers is 2231. Find the other.

• (a) 2183
• (b) 2813
• (c) 2831
• (d) 2381

(b) 2813.

To solve this problem, we can use the relationship between the HCF (GCD), LCM, and the product of two numbers. The formula is:

$$\text{Product of the two numbers} = \text{HCF} \times \text{LCM}$$

Given:

• LCM = 64699
• HCF = 97
• One of the numbers = 2231
• The other number = $x$

Using the formula:

$$2231\times x=97\times 64699$$

To find $x$:

$$x=\frac{97\times 64699}{\mathrm{2231<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"></span>}}$$

Calculating the value:

$$x=\frac{6275803}{2231}=2813$$

So, the other number is 2813.

Thus, the correct answer is (b) 2813.