HCF and LCM Questions - Solved Examples, Word Problems & MCQs

HCF and LCM are important concepts in mathematics that help solve problems related to factors, multiples, and number relationships. This page covers both topics with solved examples, word problems, MCQs, and shortcuts to help students practise effectively.

Table of Contents

• Key Terms
• HCF Questions
• LCM Questions
• HCF and LCM of 3 Numbers
• Word Problems
• MCQ Questions
• Tricks and Shortcuts
• 4 Mistakes to Avoid
• Conclusion
• Frequently Asked Questions On HCF and LCM Questions

 

Key Terms

Note: HCF is always less than or equal to the smaller number. LCM is always greater than or equal to the larger number.

HCF Questions

Q1. Find the HCF of 24 and 36.

Answer: 24 = 2 x 2 x 2 x 3 = 2³ x 3

36 = 2 x 2 x 3 x 3 = 2² x 3²

Common prime factors with lowest powers = 2² x 3 = 4 x 3 = 12

Q2. Find the HCF of 48 and 72 using the division method.

Answer: 72 / 48 = 1 remainder 24

48 / 24 = 2 remainder 0

Last divisor = HCF = 24

Q3. The HCF of two numbers is 12. One number is 60. Find the other number if their LCM is 180.

Answer: 12 x 180 = 60 x other number

Other number = (12 x 180) / 60 = 2160 / 60 = 36

Practice Questions

1. Find the HCF of 18 and 45.

2. Find the HCF of 56 and 84 using prime factorisation.

3. HCF of two numbers is 8. Their product is 960. Find their LCM.

4. Find the greatest number that divides 36, 48 and 60 exactly.

Read more:

• LCM questions
• HCF questions
• HCF and LCM

LCM Questions

Q1. Find the LCM of 12 and 18.

Answer: 12 = 2² x 3

18 = 2 x 3²

Take highest powers: 2² x 3² = 4 x 9 = 36

Q2. Find the LCM of 8, 12 and 15 using the division method.

Answer: Divide by 2: 4, 6, 15

Divide by 2: 2, 3, 15

Divide by 3: 2, 1, 5

No more common factors. Multiply all divisors and remaining numbers:

LCM = 2 x 2 x 3 x 2 x 1 x 5 = 120

Q3. LCM of two numbers is 120 and their HCF is 6. One number is 24. Find the other.

Answer: 6 x 120 = 24 x other number

Other number = 720 / 24 = 30

Practice Questions

1. Find the LCM of 15 and 20.

2. Find the LCM of 6, 8 and 12.

3. LCM of two numbers is 180 and HCF is 12. One number is 36. Find the other.

4. Find the smallest number divisible by both 14 and 21.

HCF and LCM of 3 Numbers

Q1. Find the HCF of 36, 48 and 60.

Answer: 36 = 2² x 3²

48 = 2⁴ x 3

60 = 2² x 3 x 5

Common prime factors with lowest powers = 2² x 3 = 4 x 3 = 12

Q2. Find the LCM of 12, 15 and 20.

Answer: 12 = 2² x 3

15 = 3 x 5

20 = 2² x 5

Take highest powers of all primes: 2² x 3 x 5 = 4 x 3 x 5 = 60

Q3. Find the HCF and LCM of 24, 36 and 48.

Answer: 24 = 2³ x 3

36 = 2² x 3²

48 = 2⁴ x 3

HCF = lowest powers of common factors = 2² x 3 = 12

LCM = highest powers of all prime factors = 2⁴ x 3² = 16 x 9 = 144

Practice Questions

1. Find the HCF of 30, 45 and 60.

2. Find the LCM of 8, 12 and 18.

3. Find both HCF and LCM of 15, 25 and 35.
   
   

HCF or LCM - how to decide

• Use HCF when the question asks to DIVIDE or SPLIT things into equal groups - "greatest number of students per row", "largest tile size", "maximum packets".
• Use LCM when the question asks about REPEATING events or finding the MINIMUM common value - "when will they meet again", "smallest number divisible by both", "after how many days".

Word Problems

Q1. Two bells ring at intervals of 12 minutes and 18 minutes. They ring together at 8:00 AM. When will they ring together again?

Answer: "When will they ring together" - two events repeating, asking when they coincide. That is LCM.

12 = 2² x 3

18 = 2 x 3²

LCM = 2² x 3² = 4 x 9 = 36 minutes

They will ring together again at 8:36 AM.

Q2. A teacher wants to arrange 48 boys and 60 girls in equal rows with no student left over. What is the maximum number of students in each row?

Answer: Splitting into equal groups, finding the maximum - HCF.

48 = 2⁴ x 3

60 = 2² x 3 x 5

HCF = 2² x 3 = 12

Maximum 12 students per row.

Q3. Three runners complete one lap in 30, 40 and 60 minutes respectively. They start together. After how many minutes will all three be at the starting point together again?

Answer: Three things repeating, asking when they all coincide. LCM of three numbers.

30 = 2 x 3 x 5

40 = 2³ x 5

60 = 2² x 3 x 5

LCM = 2³ x 3 x 5 = 8 x 3 x 5 = 120 minutes

Practice Questions

1. Find the largest number that divides 70, 105 and 175 exactly.
2. Two buses leave a station every 15 and 20 minutes. They leave together at 9:00 AM. When will they leave together again?
3. What is the smallest number of tiles (all the same size) that can exactly cover a floor of 6m x 9m without cutting any tile?

MCQ Questions

Frequently Asked Questions On HCF and LCM Questions

1. How to know the HCF and LCM?

Answer: To find the HCF (Highest Common Factor) and LCM (Least Common Multiple) of two numbers:

• HCF Method:

• List the factors of both numbers and find the greatest one common to both

• Or use the Euclidean algorithm (divide and take remainders)

• LCM Method:

• List multiples of both numbers and find the smallest one common to both

• Or use the formula:
  LCM(a, b) = (a × b) / HCF(a, b)

2. What is the HCF of 24 and 36?

Answer: 

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

• HCF = 12

3. What are the LCM and HCF?

Answer: 

• LCM (Least Common Multiple): The smallest number that is a multiple of two or more numbers

• HCF (Highest Common Factor): The greatest number that divides two or more numbers exactly

4. What is the HCF of 645 and 473?

 Answer: Using the Euclidean algorithm:

• 645 ÷ 473 = 1 remainder 172

• 473 ÷ 172 = 2 remainder 129

• 172 ÷ 129 = 1 remainder 43

• 129 ÷ 43 = 3 remainder 0

HCF = 43

Keep practicing HCF and LCM problems to master the concept. Explore more math lessons and solved examples with Orchids The International School to strengthen your foundation!