HCF and LCM Questions

Understanding HCF and LCM questions is a fundamental part of number theory in mathematics. These concepts help solve a variety of problems involving divisibility, time intervals, arrangement, and more.

 

Table of Contents

• HCF and LCM Formulas
  
• HCF Questions
  
• LCM Questions
  
• HCF and LCM Examples
  
• LCM and HCF for Fractions
  
• Relation Between HCF and LCM Questions
  
• Practice HCF and LCM Questions with Solutions
  
• Conclusion
  
• FAQs On HCF and LCM Questions

 

The HCF (Highest Common Factor) is the greatest number that divides two or more numbers exactly.

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

HCF and LCM formulas, and a variety of solved HCF and LCM problems-including LCM and HCF for fractions and relationship-based problems, will all be covered below.

 

HCF and LCM Formulas

To solve HCF and LCM questions, it is important to use the correct formulas.

 

HCF Formulas:

• HCF (a, b) = Product of all common prime factors

• HCF using division:
  Continue dividing until the remainder is 0. The last divisor is the HCF.

 

LCM Formulas:

• LCM (a, b) = Product of all prime factors with the highest power

• LCM (a, b) = (a × b) / HCF (a, b)
  
  

These HCF and LCM formulas are applicable to all types of HCF and LCM questions.

 

HCF Questions

Example 1:

Find the HCF of 36 and 60 using prime factorization.

36 = 2² × 3²

60 = 2² × 3 × 5

Common prime factors: 2² × 3 = 12

HCF = 12

 

Example 2:

Find the HCF of 72 and 120 using division.

120 ÷ 72 = 1, remainder = 48

72 ÷ 48 = 1, remainder = 24

48 ÷ 24 = 2, remainder = 0

HCF = 24

 

Example 3:

Find the HCF of 45, 75, and 105.

45 = 3² × 5

75 = 3 × 5²

105 = 3 × 5 × 7

Common prime factors: 3 × 5 = 15

HCF = 15

 

Example 4:

What is the HCF of 16 and 28?

16 = 2⁴

28 = 2² × 7

Common prime factor: 2² = 4

HCF = 4

 

LCM Questions

Example 1:

Find the LCM of 12 and 15.

12 = 2² × 3

15 = 3 × 5

LCM = 2² × 3 × 5 = 60

 

Example 2:

Find the LCM of 24 and 36.

24 = 2³ × 3

36 = 2² × 3²

LCM = 2³ × 3² = 72

 

Example 3:

Find the LCM of 14, 28, and 70.

14 = 2 × 7

28 = 2² × 7

70 = 2 × 5 × 7

LCM = 2² × 5 × 7 = 140

 

Example 4:

Find the LCM of 8 and 20 using the formula.

HCF of 8 and 20 = 4

LCM = (8 × 20) / 4 = 160 / 4 = 40

 

HCF and LCM Examples

Example 1:

Problem:

Two pieces of wood are 96 cm and 144 cm long. What is the greatest length that can be cut from both pieces?

Solution:

We are looking for the greatest length that can divide both 96 and 144 exactly → Use HCF.

Prime factorization:

96 = 2⁵ × 3

144 = 2⁴ × 3²

Common factors = 2⁴ × 3 = 16 × 3 = 48

Answer: 48 cm

 

Example 2:

Problem:

Two alarm clocks ring every 15 minutes and 20 minutes. If both ring together at 12 PM, when will they ring together next?

Solution:

This is a case of events repeating together → Use LCM.

15 = 3 × 5

20 = 2² × 5

LCM = 2² × 3 × 5 = 4 × 3 × 5 = 60 minutes

Next ring together at: 12 PM + 60 min = 1:00 PM

Answer: In 60 minutes or at 1:00 PM

Example 3:

Problem:

A gardener has 90 rose plants and 150 jasmine plants. He wants to arrange them in equal rows with no plant left over. Find the maximum number of rows possible.

Solution:

Maximum number of rows → Use HCF.

Prime factorization:

90 = 2 × 3² × 5

150 = 2 × 3 × 5²

Common prime factors = 2 × 3 × 5 = 30

Answer: 30 rows

Example 4:

Problem:

Find the smallest number that is divisible by 8, 9, and 12.

Solution:

Smallest number divisible by all → Use LCM.

8 = 2³

9 = 3²

12 = 2² × 3

LCM = 2³ × 3² = 8 × 9 = 72

Answer: 72

 

LCM and HCF for Fractions

To solve LCM and HCF for fractions, use these formulas:

HCF of fractions = HCF of numerators / LCM of denominators

LCM of fractions = LCM of numerators / HCF of denominators

 

Example 1:

Find the HCF and LCM of 2/3 and 4/5.

HCF of 2 and 4 = 2

LCM of 3 and 5 = 15

HCF = 2 / 15

LCM of 2 and 4 = 4

HCF of 3 and 5 = 1

LCM = 4 / 1 = 4

 

Example 2:

Find the HCF and LCM of 3/7 and 6/14.

HCF(3,6) = 3, LCM(7,14) = 14

HCF = 3 / 14

LCM(3,6) = 6, HCF(7,14) = 7

LCM = 6 / 7

 

Example 3:

Find the HCF and LCM of:
5/6, 10/9, and 15/12

Step 1: Numerators = 5, 10, 15
Step 2: Denominators = 6, 9, 12

HCF of numerators = HCF(5, 10, 15) = 5
LCM of denominators = LCM(6, 9, 12) = 36

HCF = 5 / 36

LCM of numerators = LCM(5, 10, 15) = 30
HCF of denominators = HCF(6, 9, 12) = 3

LCM = 30 / 3 = 10

 

Example 4:

Find the HCF and LCM of:
4/7, 8/21

Step 1: Numerators = 4, 8
Step 2: Denominators = 7, 21

HCF of numerators = HCF(4, 8) = 4
LCM of denominators = LCM(7, 21) = 21

HCF = 4 / 21

LCM of numerators = LCM(4, 8) = 8
HCF of denominators = HCF(7, 21) = 7

LCM = 8 / 7

 

Example 5:

Find the HCF and LCM of:
2/5, 3/10, 4/15

Step 1: Numerators = 2, 3, 4
Step 2: Denominators = 5, 10, 15

HCF of numerators = HCF(2, 3, 4) = 1
LCM of denominators = LCM(5, 10, 15) = 30

HCF = 1 / 30

LCM of numerators = LCM(2, 3, 4) = 12
HCF of denominators = HCF(5, 10, 15) = 5

LCM = 12 / 5

 

Relation Between HCF and LCM Questions

The product of two numbers = HCF × LCM

This relation is useful in many HCF and LCM questions.

 

Question 1: If the HCF of two numbers is 12 and their LCM is 180, find the product of the numbers.

Solution:

We use the formula:

HCF × LCM = Product of the two numbers

= 12 × 180 = 2160

Answer: Product of the two numbers = 2160

 

Question 2: If the product of two numbers is 960 and their HCF is 12, find their LCM.

Solution:

HCF × LCM = Product of numbers

12 × LCM = 960

LCM = 960 ÷ 12 = 80

Answer: LCM = 80

Question 3: The LCM of two numbers is 120 and the HCF is 6. If one number is 30, find the other number.

Solution:

Use the relation:

Product of numbers = HCF × LCM = 6 × 120 = 720

Other number = 720 ÷ 30 = 24

Answer: The other number is 24

 

Question 4: Two numbers have HCF = 4 and LCM = 240. If one number is 60, find the other.

Solution:

Product of numbers = HCF × LCM = 4 × 240 = 960

Other number = 960 ÷ 60 = 16

Answer: The other number is 16

 

Question 5: The two numbers are 36 and 60. Verify that HCF × LCM = Product of the numbers.

Solution:

HCF(36, 60) = 12

LCM(36, 60) = 180

HCF × LCM = 12 × 180 = 2160

36 × 60 = 2160

Verified.

Answer: HCF × LCM = Product of numbers = 2160

 

Question 6: The HCF and LCM of two numbers are 11 and 770 respectively. If one number is 55, find the other.

Solution:

Product = HCF × LCM = 11 × 770 = 8470

Other number = 8470 ÷ 55 = 154

Answer: The other number is 154

 

Practice HCF and LCM Questions with Solutions

1. Find the HCF and LCM of 45 and 60.

2. Find the LCM of 5, 10, and 15.

3. Find the HCF of 28 and 42.

4. Find the smallest number divisible by 6, 8, and 12.

5. Find the HCF and LCM of 3/4 and 9/10.

6. Two buses leave a station at the same time. One returns every 24 minutes, the other every 36 minutes. When will they meet again?
   LCM = 72 minutes

7. A rectangular tile of length 18 cm and width 24 cm is to be used to cover a floor. Find the largest size of square tile that can be used.
   HCF = 6 cm

 

Conclusion

You can solve mathematical problems involving divisibility, arrangement, and multiples more effectively if you comprehend and practice hcf and lcm questions. Verify all HCF and LCM formulas, solve HCF problems, try LCM problems, and have a clear understanding of how HCF and LCM questions relate to one another.

Learn the formulas for LCM and HCF for fractions, and practice a wide range of HCF and LCM examples on a regular basis.

 

Related Links

HCF and LCM -  Quickly grasp the basics with simple explanations and examples

HCF (Highest Common Factor) - Understand with step-by-step explanations

Integers Questions - Practice and strengthen your basics with solved problems

 

Frequently Asked Questions (FAQs) On HCF and LCM Questions

1. How to know the HCF and LCM?

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?

• 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?

• 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?

 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!