PAGE 5

HCF & LCM

HIGHEST COMMON FACTOR

01. Introduction to HCF

HCF stands for Highest Common Factor. It is the greatest number that divides two or more given numbers exactly without leaving any remainder.

HCF = Highest Common Factor

Example:

Find the HCF of 12 and 18.

Solution:

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

Common factors: 1, 2, 3, 6
Highest common factor = 6

HCF of 12 and 18 = 6

02. Methods of Finding HCF

15.1 Method 1 — Listing Factors

In this method, write all the factors of each number and select the greatest factor common to all numbers.

Example: Find 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

Common factors: 1, 2, 3, 4, 6, 12
Therefore, HCF = 12.

15.2 Method 2 — Prime Factorisation

In prime factorisation, express each number as a product of prime numbers. The product of the common prime factors with the smallest powers gives the HCF.

Example: Find HCF of 36 and 48

36 = 2 × 2 × 3 × 3
48 = 2 × 2 × 2 × 2 × 3

Common prime factors: 2 × 2 × 3

Therefore: HCF = 12

03. HCF by Division Method

The division method is especially useful for large numbers. Divide the larger number by the smaller number. Then divide the previous divisor by the remainder. Continue until the remainder becomes zero.

Example: HCF of 48 and 18

48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0

Since the last non-zero remainder is 6,
HCF = 6

04. Important Points about HCF

HCF is always a factor of each given number.
HCF cannot be greater than the smallest given number.
If two numbers are co-prime, their HCF is 1.
The HCF of a number and 1 is 1.

PAGE 6

HCF & LCM

LEAST COMMON MULTIPLE

05. Introduction to LCM

LCM stands for Least Common Multiple. It is the smallest positive number that is exactly divisible by all the given numbers.

LCM = Least Common Multiple

Example:

Find the LCM of 6 and 8.

Solution:

Multiples of 6: 6, 12, 18, 24, 30, 36, ...
Multiples of 8: 8, 16, 24, 32, 40, ...

First common multiple = 24

LCM of 6 and 8 = 24

06. Methods of Finding LCM

06.1 Method 1 — Listing Multiples

Write the multiples of each number and identify the smallest number common to all the lists.

Example: Find LCM of 4 and 6

Multiples of 4: 4, 8, 12, 16, 20, ...
Multiples of 6: 6, 12, 18, 24, ...

The smallest common multiple is 12.

06.2 Method 2 — Prime Factorisation

Express each number as a product of prime factors. Take every prime factor with its greatest power. Their product gives the LCM.

Example: Find LCM of 12 and 18

12 = 2 × 2 × 3
18 = 2 × 3 × 3

Greatest powers: 2² and 3²

LCM = 2² × 3²
LCM = 4 × 9
LCM = 36

07. LCM by Division Method

In the division method, divide the given numbers by suitable prime numbers until all numbers become 1. The product of all divisors gives the LCM.

Example: Find LCM of 8, 12 and 18

Divisor	8	12	18
2	4	6	9
2	2	3	9
2	1	3	9
3	1	1	3
3	1	1	1

LCM = 2 × 2 × 2 × 3 × 3
LCM = 72

08. Important Points about LCM

LCM is always a multiple of each given number.
LCM is never smaller than the greatest given number.
The LCM of two co-prime numbers is their product.
The LCM of 1 and any number is that number.

PAGE 7

HCF & LCM

IMPORTANT FORMULAS & EXAMPLES

09. Relation Between HCF and LCM

For two positive integers, the product of the HCF and LCM is equal to the product of the two numbers.

HCF × LCM = First Number × Second Number

Example 1:

Find the LCM of 12 and 18 if their HCF is 6.

Solution:

HCF × LCM = 12 × 18
6 × LCM = 216
LCM = 216 ÷ 6
LCM = 36

Example 2:

The HCF of two numbers is 8 and their LCM is 240. If one number is 40, find the other number.

Solution:

HCF × LCM = First Number × Second Number

8 × 240 = 40 × Second Number

1920 = 40 × Second Number

Second Number = 1920 ÷ 40
Second Number = 48

10. HCF and LCM of Three Numbers

For three or more numbers, HCF can be found by taking the common prime factors with their smallest powers. LCM is found by taking all prime factors with their greatest powers.

Example:

Find the HCF and LCM of 12, 18 and 24.

Prime Factorisation:

12 = 2² × 3
18 = 2 × 3²
24 = 2³ × 3

HCF:
Common factors with smallest powers:
2¹ × 3¹ = 6

LCM:
Greatest powers:
2³ × 3² = 8 × 9 = 72

11. HCF & LCM Quick Reference

Concept	Meaning	Key Idea
HCF	Highest Common Factor	Greatest common divisor
LCM	Least Common Multiple	Smallest common multiple
HCF Method	Prime Factorisation	Take smallest powers
LCM Method	Prime Factorisation	Take greatest powers
Relation	For two numbers	HCF × LCM = Product

12. Useful Shortcuts

Shortcut Rules

1. If two numbers are co-prime:
HCF = 1
LCM = Product of the numbers.

2. If one number divides another number exactly:
HCF = Smaller number
LCM = Larger number.

3. For two numbers:
LCM = (First Number × Second Number) ÷ HCF

13. Solved Mixed Example

Find the HCF and LCM of 20 and 30.

Solution:

20 = 2² × 5
30 = 2 × 3 × 5

HCF = 2 × 5 = 10
LCM = 2² × 3 × 5 = 60

Verification:
HCF × LCM = 10 × 60 = 600
20 × 30 = 600

Therefore, the answer is correct.

HCF = Smallest Powers | LCM = Greatest Powers

```