HCF and LCM Questions

HCF (Highest Common Factor) and LCM (Least Common Multiple) are two of the most important concepts in mathematics and number theory. They form the foundation for solving problems related to divisibility, multiples, time intervals, arrangement of objects, and fractions. Understanding HCF and LCM not only strengthens basic arithmetic skills but also helps in solving advanced mathematical questions with ease. In this guide, we will explore formulas, solved examples, and word problems.

 

Table of Contents

• How to Calculate the HCF?
• HCF Questions
• How to Calculate the LCM?
• 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.

To solve HCF and LCM questions, it is important to understand the correct method. 

 

How to Calculate the HCF

In order to solve HCF questions, we can apply different methods of calculating HCF. Below are the three methods used most commonly to find HCF:

 

HCF Questions

Prime Factorization Method

Rule: Take common prime factors (with the lowest powers) and multiply them.

Example 1: Find HCF(60, 90)

Step 1: Prime factorize each number.

• 60 = 2² × 3 × 5

• 90 = 2 × 3² × 5
  
  

Step 2: Take common primes with the lowest powers.

• Common = 2¹ × 3¹ × 5¹

Step 3: Multiply the common factors.

• HCF = 2 × 3 × 5 = 30

Answer: 30

 

Example 2: Find HCF(84, 126, 210)

Step 1: Prime factorize.

• 84 = 2² × 3 × 7

• 126 = 2 × 3² × 7

• 210 = 2 × 3 × 5 × 7
  
  

Step 2: Take primes common to all three with lowest powers.

• Common = 2¹ × 3¹ × 7¹
  
  

Step 3: Multiply.

• HCF = 2 × 3 × 7 = 42

Answer: 42

 

Division Method 

 Rule: Successively divide; when the remainder becomes 0, the last divisor is the HCF.

Example 1: Find HCF(252, 105)

Step 1: 252 ÷ 105 = 2 remainder 42
Step 2: 105 ÷ 42 = 2 remainder 21
Step 3: 42 ÷ 21 = 2 remainder 0 → stop

Conclusion: Last non-zero divisor = 21

Answer: 21

 

Example 2: Find HCF(96, 144, 180)

Step 1: First, find HCF(96, 144).

• 144 ÷ 96 = 1 r 48
  
  

• 96 ÷ 48 = 2 r 0 → HCF(96, 144) = 48
  
  

Step 2: Now find HCF(48, 180).

• 180 ÷ 48 = 3 r 36

• 48 ÷ 36 = 1 r 12

• 36 ÷ 12 = 3 r 0 → HCF = 12
  
  

Answer: 12

 

Listing Method

Rule: List all factors of each number, find the common ones, and pick the greatest.

Example 1: Find HCF(18, 30)

Step 1: Factors of 18 = 1, 2, 3, 6, 9, 18
Step 2: Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 30
Step 3: Common factors = 1, 2, 3, 6 → Greatest = 6

Answer: 6

 

Example 2: Find HCF(24, 36, 60)

Step 1:

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

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

• Factors of 60 = 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  
  

Step 2: Common factors = 1, 2, 3, 4, 6, 12 → Greatest = 12

Answer: 12

 

How to Calculate the LCM?

To solve LCM questions, we can apply various methods for calculating the LCM. Below are the three methods used most commonly to find LCM:

The above-stated methods of finding LCM will make it easy for you to solve all types of LCM questions, whether small or large numbers.

 

LCM Questions

Prime Factorization Method

Rule: Take all prime factors (with the highest powers) and multiply them.

Example 1: Find LCM(12, 18)

Step 1: Prime factorize each number.
12 = 2 × 2 × 3
18 = 2 × 3 × 3

Step 2: Take all primes with the highest powers.
= 2 × 2 × 3 × 3

Step 3: Multiply.
LCM = 36

Answer: 36

 

Example 2: Find LCM(15, 20, 30)

Step 1: Prime factorize.
15 = 3 × 5
20 = 2 × 2 × 5
30 = 2 × 3 × 5

Step 2: Take primes with highest powers.
= 2 × 2 × 3 × 5

Step 3: Multiply.
LCM = 60

Answer: 60

 

Division Method

Rule: Divide the numbers by prime numbers step by step until all become 1. Multiply all the divisors to get the LCM.

Example 1: Find LCM(12, 18, 24)

Step 1: 12, 18, 24 ÷ 2 = 6, 9, 12
Step 2: 6, 9, 12 ÷ 2 = 3, 9, 6
Step 3: 3, 9, 6 ÷ 3 = 1, 3, 2
Step 4: 1, 3, 2 ÷ 3 = 1, 1, 2
Step 5: 1, 1, 2 ÷ 2 = 1, 1, 1

Conclusion: Multiply all divisors = 2 × 2 × 3 × 3 × 2 = 72

Answer: 72

 

Example 2: Find LCM(16, 20)

Step 1: 16, 20 ÷ 2 = 8, 10
Step 2: 8, 10 ÷ 2 = 4, 5
Step 3: 4, 5 ÷ 2 = 2, 5
Step 4: 2, 5 ÷ 2 = 1, 5
Step 5: 1, 5 ÷ 5 = 1, 1

Conclusion: Multiply all divisors = 2 × 2 × 2 × 2 × 5 = 80

Answer: 80

 

Listing Method

Rule: List the multiples of each number and pick the first (lowest) common one.

Example 1: Find LCM(6, 8)

Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48 …
Multiples of 8 = 8, 16, 24, 32, 40, 48 …

Common multiples = 24, 48 …
Lowest = 24

Answer: 24

 

Example 2: Find LCM(9, 12, 15)

Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 …
Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96 …
Multiples of 15 = 15, 30, 45, 60, 75, 90 …

Common multiples = 180, 360 …
Lowest = 180

Answer: 180

 

Solved Word Problems on HCF and LCM

1. Determine the largest number that perfectly divides 60, 75, and 90.

Solution:
We have to find the HCF of 60, 75 and 90.

Prime factorisation of 60 = 2 × 2 × 3 × 5

Prime factorisation of 75 = 3 × 5 × 5

Prime factorisation of 90 = 2 × 3 × 3 × 5

The common prime factors = 3 × 5 = 15

Therefore, the number required = 15

Answer: 15

 

2. Determine the smallest number which, when divided by 12, 15, and 20 leaves a remainder of 5 in all cases.

Solution:
Step 1: The number leaves equal remainder (5), so subtract 5 from the number.
Thus, the number - 5 is divisible by 12, 15 and 20.

Step 2: Determine LCM (12, 15, 20)

Prime factors of 12 = 2 × 2 × 3

Prime factors of 15 = 3 × 5

Prime factors of 20 = 2 × 2 × 5

LCM = 2 × 2 × 3 × 5 = 60

Step 3: Required number = 60 + 5 = 65

Answer: 65

 

3. The traffic lights at three road crossings change after 48 seconds, 72 seconds and 108 seconds. After how much time will they change together again?

Solution:
We have to determine the LCM (48, 72, 108)

48 = 2⁴ × 3

72 = 2³ × 3²

108 = 2² × 3³

LCM = 2⁴ × 3³ = 432 seconds

Convert to minutes = 432 ÷ 60 = 7 minutes 12 seconds

Answer: In 7 minutes 12 seconds

 

4. Determine the largest length of a rope that can measure exactly the lengths 72 m, 108 m and 210 m.

Solution:
We must determine the HCF (72, 108, 210)

72 = 2³ × 3²

108 = 2² × 3³

210 = 2 × 3 × 5 × 7

Common factors = 2 × 3 = 6

Therefore, the greatest length of rope = 6 m

Answer: 6 m

 

5. Determine the smallest number that is divisible by 24, 36 and 40.

Solution:
We require the LCM (24, 36, 40)

24 = 2³ × 3

36 = 2² × 3²

40 = 2³ × 5

LCM = 2³ × 3² × 5 = 360

Answer: 360

 

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

 

Question 1: Find the LCM of 2/3 and 4/9.

Solution:

Step 1: Find LCM of numerators 2 and 4 → LCM(2, 4) = 4

Step 2: Find HCF of denominators 3 and 9 → HCF(3, 9) = 3

Step 3: Apply formula → LCM = LCM of numerators / HCF of denominators
LCM = 4 / 3

Answer: LCM of 2/3 and 4/9 = 4/3

 

Question 2:

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

Solution:

Step 1: Determine HCF of numerators 3 and 9 → HCF(3, 9) = 3

Step 2: Determine LCM of denominators 4 and 10 → LCM(4, 10) = 20

Step 3: Use formula → HCF = HCF of numerators / LCM of denominators
HCF = 3 / 20

Answer: HCF of 3/4 and 9/10 = 3/20

 

Question 3:

Determine the LCM of 5/12 and 7/18.

Solution:

Step 1: Determine LCM of numerators 5 and 7 → LCM(5, 7) = 35

Step 2: Determine HCF of denominators 12 and 18 → HCF(12, 18) = 6

Step 3: Use formula → LCM = LCM of numerators / HCF of denominators
LCM = 35 / 6

Answer: LCM of 5/12 and 7/18 = 35/6

 

Question 4:

Determine the HCF of 16/45 and 24/75.

Solution:

Step 1: Determine HCF of numerators 16 and 24 → HCF(16, 24) = 8

Step 2: Determine LCM of denominators 45 and 75 → LCM(45, 75) = 225

Step 3: Use the formula → HCF = HCF of numerators / LCM of denominators
HCF = 8 / 225

Answer: HCF of 16/45 and 24/75 = 8/225

 

Question 5:

Determine the LCM of 11/14 and 22/35.

Solution:

Step 1: Determine LCM of numerators 11 and 22 → LCM(11, 22) = 22

Step 2: Determine HCF of denominators 14 and 35 → HCF(14, 35) = 7

Step 3: Use formula → LCM = LCM of numerators / HCF of denominators
LCM = 22 / 7

Answer: LCM of 11/14 and 22/35 = 22/7

 

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.

 

Frequently Asked Questions On HCF and LCM Questions

1. How to know the HCF and LCM?

Ans: 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?

Ans:

• 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?

Ans:

• 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?

 Ans: 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!