# HCF and LCM of Numbers

### Table of Contents

## Introduction to HCF and LCM

#### Highest Common Factor

- HCF stands for
**Highest Common Factor**. - It is the
**largest positive integer**that divides two or more numbers without leaving a remainder.

**How to Find the HCF**

There are two main methods to find the HCF: the **prime factorization method** and the **division method**.

#### Prime Factorization Method

**Step 1:**Find the prime factors of each number.**Step 2:**Identify the common prime factors.**Step 3:**Multiply these common prime factors to get the HCF.

**Example: Finding HCF of 36 and 48 using Prime Factorization Method**

- Prime factors of 36 = 2 × 2 × 3 × 3 = 2² × 3²
- Prime factors of 48 = 2 × 2 × 2 × 2 × 3 = 2^4 × 3¹
- Common prime factors = 2² × 3
- HCF = 2² × 3 = 4 × 3 = 12

#### Division Method

**Step 1:**Divide the larger number by the smaller number.**Step 2:**Replace the larger number with the remainder and repeat the process.**Step 3:**Continue this process until the remainder is zero. The divisor at this stage is the HCF.

**Example: Finding HCF of 56 and 98 using Division Method**

- 98 ÷ 56 = 1 remainder 42
- 56 ÷ 42 = 1 remainder 14
- 42 ÷ 14 = 3 remainder 0
- HCF = 14

#### Least Common Multiple

**LCM**stands for**Least Common Multiple**.- It is defined as the
**smallest positive integer**that is divisible by two or more numbers without leaving a remainder.

**How to Find LCM:**

#### Prime Factorization Method

**Break down each number**into its**prime factors**.**Multiply**the highest power of each prime factor to get the LCM.

**Examples: ****Using Prime Factorization Method:**

Find the LCM of **6 and 8**.

Prime factors of 6 = 2 x 3

Prime factors of 8 = 2 x 2 x 2

LCM = 2 x 2 x 2 x 3 = 24

#### Division Method

**Write down**the numbers you want to find the LCM for.**Divide**each number by the**smallest prime number**starting from 2 until all numbers cannot be divided further.**Multiply**the prime numbers and the quotients together to get the LCM.

**Examples: ****Using Division Method:**

Find the LCM of **12 and 15**.

Divide 12 by 2 = 6

Divide 6 by 2 = 3

Divide 3 by 3 = 1

Divide 15 by 3 = 5

LCM = 2 x 2 x 3 x 5 = 60

## Problems of HCF and LCM

**1. Problem: **Find the HCF and LCM of 18 and 24.

**Solution:****Step 1:**

List the factors of 18: 1, 2, 3, 6, 9, 18

List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

**Step 2:**

The common factors are: 1, 2, 3, 6**HCF** = 6

**Step 3:**

The LCM can be found using the formula:

$\text{LCM}=\frac{\text{Productofthenumbers}}{\text{HCF}}$

$\text{LCM}=\frac{18\times 24}{6}$

$\text{LCM}=\frac{432}{6}$

$\text{LCM}=72$

**HCF:** 6**LCM:** 72

**2. Problem: **If the HCF of two numbers is 8 and one of the numbers is 32, find the other number.

**Solution:****Step 1:**

Using the formula:

$\text{HCF}=\frac{\text{Productofthenumbers}}{\text{LCM}}$

$\text{LCM}=\frac{32}{8}$

$\text{LCM}=4$

**Step 2:**

The LCM of 32 and the unknown number is 4.

The other number =

$32\xf74=8$

**Other number:** 8

**3. Problem: **Find the HCF and LCM of 15 and 25.

**Solution:****Step 1:**

List the factors of 15: 1, 3, 5, 15

List the factors of 25: 1, 5, 25

**Step 2: **The common factors are: 1, 5**HCF** = 5

**Step 3:**

$\text{LCM}=\frac{15\times 25}{5}$

$\text{LCM}=\frac{375}{5}$

$\text{LCM}=75$

**HCF:** 5**LCM:** 75

**4. Problem: **If the HCF of two numbers is 7 and their LCM is 42, what is the product of the numbers?

**Solution:****Step 1:**

Using the formula:

$\text{Productofthenumbers}=\text{HCF}\times \text{LCM}$

$\text{Productofthenumbers}=7\times 42$

$\text{Productofthenumbers}=294$

**Product of the numbers:** 294

**5. Problem: **The product of two numbers is 120 and their LCM is 40. Find their HCF.

**Solution:****Step 1:**

Using the formula:

$\text{HCF}=\frac{\text{Productofthenumbers}}{\text{LCM}}$

$\text{HCF}=\frac{120}{40}$

$\text{HCF}=3$

**HCF:** 3

**6. Problem: **Find the HCF and LCM of 12 and 18.

**Solution:****Step 1:**

List the factors of 12: 1, 2, 3, 4, 6, 12

List the factors of 18: 1, 2, 3, 6, 9, 18

**Step 2: **The common factors are: 1, 2, 3, 6**HCF** = 6

**Step 3:**

$\text{LCM}=\frac{12\times 18}{6}$

$\text{LCM}=\frac{216}{6}$

$\text{LCM}=36$

**HCF:** 6**LCM:** 36

**7. Problem: **If the HCF of two numbers is 9 and their LCM is 72, what is the product of the numbers?

**Solution:****Step 1:**

Using the formula:

$\text{Productofthenumbers}=\text{HCF}\times \text{LCM}$

$\text{Productofthenumbers}=9\times 72$

$\text{Productofthenumbers}=648$

**Product of the numbers:** 648

**8. Problem: **Find the HCF and LCM of 20 and 25.

**Solution:****Step 1:**

List the factors of 20: 1, 2, 4, 5, 10, 20

List the factors of 25: 1, 5, 25

**Step 2:**

The common factors are: 1, 5**HCF** = 5

**Step 3:**

$\text{LCM}=\frac{20\times 25}{5}$

$\text{LCM}=\frac{500}{5}$

$\text{LCM}=100$

**HCF:** 5**LCM:** 100

**9. Problem: **The product of two numbers is 200 and their LCM is 50. Find their HCF.

**Solution:****Step 1:**

Using the formula:

$\text{HCF}=\frac{\text{Productofthenumbers}}{\text{LCM}}$

$\text{HCF}=\frac{200}{50}$

$\text{HCF}=4$

**HCF:** 4

**10. Problem: **Find the HCF and LCM of 30 and 45.

**Solution:****Step 1:**

List the factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

List the factors of 45: 1, 3, 5, 9, 15, 45

**Step 2:**

The common factors are: 1, 3, 5, 15**HCF** = 15

**Step 3:**

$\text{LCM}=\frac{30\times 45}{15}$

$\text{LCM}=\frac{1350}{15}$

$\text{LCM}=90$

**HCF:** 15**LCM:** 90

Certainly, here are 10 more problems involving HCF and LCM, solved step-by-step:

**11. Problem: **Find the HCF and LCM of 28 and 42.

**Solution:****Step 1:**

List the factors of 28: 1, 2, 4, 7, 14, 28

List the factors of 42: 1, 2, 3, 6, 7, 14, 21, 42

**Step 2:**

The common factors are: 1, 2, 7, 14**HCF** = 14

**Step 3:**

$\text{LCM}=\frac{28\times 42}{14}$

$\text{LCM}=\frac{1176}{14}$

$\text{LCM}=84$

**HCF:** 14**LCM:** 84

**12. Problem: **If the HCF of two numbers is 12 and one of the numbers is 60, find the other number.

**Solution:****Step 1:**

Using the formula:

$\text{HCF}=\frac{\text{Productofthenumbers}}{\text{LCM}}$

$\text{LCM}=\frac{60}{12}$

$\text{LCM}=5$

**Step 2:**

The LCM of 60 and the unknown number is 5.

The other number = $60\xf75=12$

**Other number:** 12

**13. Problem:**

Find the HCF and LCM of 16 and 24.

**Solution:****Step 1:**

List the factors of 16: 1, 2, 4, 8, 16

List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

**Step 2:**

The common factors are: 1, 2, 4, 8**HCF** = 8

**Step 3:**

$\text{LCM}=\frac{16\times 24}{8}$

$\text{LCM}=\frac{384}{8}$

$\text{LCM}=48$

**HCF:** 8**LCM:** 48

**14. Problem:**

If the HCF of two numbers is 15 and their LCM is 180, what is the product of the numbers?

**Solution:****Step 1:**

Using the formula:

$\text{Productofthenumbers}=\text{HCF}\times \text{LCM}$

$\text{Productofthenumbers}=15\times 180$

$\text{Productofthenumbers}=2700$

**Product of the numbers:** 2700

**15. Problem:**

Find the HCF and LCM of 36 and 48.

**Solution:****Step 1:**

List the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

**Step 2:**

The common factors are: 1, 2, 3, 4, 6, 12**HCF** = 12

**Step 3:**

$\text{LCM}=\frac{36\times 48}{12}$

$\text{LCM}=\frac{1728}{12}$

$\text{LCM}=144$

**HCF:** 12**LCM:** 144

**16. Problem:**

If the HCF of two numbers is 20 and one of the numbers is 100, find the other number.

**Solution:****Step 1:**

Using the formula:

$\text{HCF}=\frac{\text{Productofthenumbers}}{\text{LCM}}$

$\text{LCM}=\frac{100}{20}$

$\text{LCM}=5$

**Step 2:**

The LCM of 100 and the unknown number is 5.

The other number = $100\xf75=20$

**Other number:** 20

**17. Problem:**

Find the HCF and LCM of 50 and 75.

**Solution:****Step 1:**

List the factors of 50: 1, 2, 5, 10, 25, 50

List the factors of 75: 1, 3, 5, 15, 25, 75

**Step 2:**

The common factors are: 1, 5, 25**HCF** = 25

**Step 3:**

$\text{LCM}=\frac{50\times 75}{25}$

$\text{LCM}=\frac{3750}{25}$

$\text{LCM}=150$

**HCF:** 25**LCM:** 150

**18. Problem:**

If the HCF of two numbers is 10 and their LCM is 90, what is the product of the numbers?

**Solution:****Step 1:**

Using the formula:

$\text{Productofthenumbers}=\text{HCF}\times \text{LCM}$

$\text{Productofthenumbers}=10\times 90$

$\text{Productofthenumbers}=900$

**Product of the numbers:** 900

**19. Problem:**

Find the HCF and LCM of 32 and 48.

**Solution:****Step 1:**

List the factors of 32: 1, 2, 4, 8, 16, 32

List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

**Step 2:**

The common factors are: 1, 2, 4, 8, 16**HCF** = 16

**Step 3:**

$\text{LCM}=\frac{32\times 48}{16}$

$\text{LCM}=\frac{1536}{16}$

$\text{LCM}=96$

**HCF:** 16**LCM:** 96

**20. Problem:**

If the HCF of two numbers is 14 and one of the numbers is 56, find the other number.

**Solution:****Step 1:**

Using the formula:

$\text{HCF}=\frac{\text{Productofthenumbers}}{\text{LCM}}$

$\text{LCM}=\frac{56}{14}$

$\text{LCM}=4$

**Step 2:**

The LCM of 56 and the unknown number is 4.

The other number = $56\xf74=14$

**Other number:** 14

## Related Links

## MCQ’s

**1. What does HCF stand for?**

A) Highest Common Factor

B) Highest Common Figure

C) High Common Factor

D) Highest Common Fraction**Answer: A) Highest Common Factor**

**2. Which of the following is the smallest common multiple of two or more numbers?**

A) HCF

B) LCM

C) GCD

D) Prime number**Answer: B) LCM**

**3. What is the HCF of 12 and 18?**

A) 3

B) 4

C) 6

D) 12**Answer: A) 3**

**4. What is the LCM of 6 and 8?**

A) 24

B) 12

C) 48

D) 18**Answer: A) 24**

**5. If the HCF of two numbers is 5 and their LCM is 30, what is the product of the numbers?**

A) 100

B) 120

C) 150

D) 125**Answer: C) 150**

**6. What is the HCF of 16 and 24?**

A) 4

B) 6

C) 8

D) 12**Answer: A) 4**

**7. What is the LCM of 9 and 15?**

A) 30

B) 45

C) 18

D) 25**Answer: B) 45**

**8. If the HCF of two numbers is 7 and one of the numbers is 28, what is the other number?**

A) 14

B) 21

C) 35

D) 42**Answer: A) 14**

**9. What is the HCF of 20 and 25?**

A) 5

B) 10

C) 15

D) 20**Answer: A) 5**

**10. If the LCM of two numbers is 60 and one of the numbers is 12, what is the other number?**

A) 5

B) 10

C) 15

D) 20**Answer: B) 10**

**11. What is the HCF of 8 and 12?**

A) 2

B) 3

C) 4

D) 6**Answer: A) 2**

**12. What is the LCM of 5 and 7?**

A) 20

B) 25

C) 30

D) 35**Answer: C) 30**

**13. If the HCF of two numbers is 9 and their LCM is 72, what is the product of the numbers?**

A) 648

B) 720

C) 6480

D) 729**Answer: A) 648**

**14. What is the HCF of 15 and 25?**

A) 3

B) 5

C) 10

D) 15**Answer: B) 5**

**15. What is the LCM of 10 and 12?**

A) 20

B) 30

C) 40

D) 50**Answer: B) 30**

**16. If the HCF of two numbers is 8 and one of the numbers is 32, what is the other number?**

A) 16

B) 24

C) 40

D) 48**Answer: A) 16**

**17. What is the HCF of 18 and 24?**

A) 4

B) 6

C) 8

D) 12**Answer: C) 6**

**18. What is the LCM of 4 and 5?**

A) 10

B) 15

C) 20

D) 25**Answer: C) 20**

**19. If the HCF of two numbers is 11 and their LCM is 121, what is the product of the numbers?**

A) 121

B) 132

C) 110

D) 1210**Answer: D) 1210**

**20. What is the LCM of 3 and 9?**

A) 6

B) 9

C) 12

D) 18**Answer: D) 18**