HCF and LCM of Numbers

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:

$LCM = \frac{Product of the numbers}{HCF}$

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

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

$LCM = 72$

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:

$HCF = \frac{Product of the numbers}{LCM}$

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

$LCM = 4$

Step 2:
The LCM of 32 and the unknown number is 4.
The other number =

$32 \div 4 = 8$

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:

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

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

$LCM = 75$

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:

$Product of the numbers = HCF \times LCM$

$Product of the numbers = 7 \times 42$

$Product of the numbers = 294$

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:

$HCF = \frac{Product of the numbers}{LCM}$

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

$HCF = 3$

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:

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

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

$LCM = 36$

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:

$Product of the numbers = HCF \times LCM$

$Product of the numbers = 9 \times 72$

$Product of the numbers = 648$

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:

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

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

$LCM = 100$

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:

$HCF = \frac{Product of the numbers}{LCM}$

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

$HCF = 4$

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:

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

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

$LCM = 90$

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:

$LCM = \frac{28 \times 42}{14}$
$LCM = \frac{1176}{14}$
$LCM = 84$

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:
$HCF = \frac{Product of the numbers}{LCM}$
$LCM = \frac{60}{12}$
$LCM = 5$

Step 2:
The LCM of 60 and the unknown number is 5.
The other number = $60 \div 5 = 12$

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:
$LCM = \frac{16 \times 24}{8}$
$LCM = \frac{384}{8}$
$LCM = 48$

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:
$Product of the numbers = HCF \times LCM$
$Product of the numbers = 15 \times 180$
$Product of the numbers = 2700$

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:
$LCM = \frac{36 \times 48}{12}$
$LCM = \frac{1728}{12}$
$LCM = 144$

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:
$HCF = \frac{Product of the numbers}{LCM}$
$LCM = \frac{100}{20}$
$LCM = 5$

Step 2:
The LCM of 100 and the unknown number is 5.
The other number = $100 \div 5 = 20$

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:
$LCM = \frac{50 \times 75}{25}$
$LCM = \frac{3750}{25}$
$LCM = 150$

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:
$Product of the numbers = HCF \times LCM$
$Product of the numbers = 10 \times 90$
$Product of the numbers = 900$

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:
$LCM = \frac{32 \times 48}{16}$
$LCM = \frac{1536}{16}$
$LCM = 96$

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:
$HCF = \frac{Product of the numbers}{LCM}$
$LCM = \frac{56}{14}$
$LCM = 4$

Step 2:
The LCM of 56 and the unknown number is 4.
The other number = $56 \div 4 = 14$

Other number: 14

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

HCF and LCM of Numbers

Table of Contents

Introduction to HCF and LCM

Highest Common Factor

Prime Factorization Method

Division Method

Least Common Multiple

Prime Factorization Method

Division Method

Problems of HCF and LCM

Related Links

MCQ’s

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