HCF AND LCM

⭐ Two Important Concepts – Let's Understand! ⭐
1. HCF (Highest Common Factor)

HCF is the greatest number that divides two or more numbers exactly.

HCF tells us the biggest common factor.

Example:

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 of these is 6.

So, HCF of 12 and 18 = 6
WHAT IS A FACTOR?

A factor is a number that divides another number exactly with no remainder.


Example:

Factors of 15 = 1, 3, 5, 15

because 15 ÷ 1 = 15,

15 ÷ 3 = 5,

15 ÷ 5 = 3,

15 ÷ 15 = 1

WHAT IS A MULTIPLE?

A multiple is the product of a number and any whole number.


Example:

Multiples of 4 = 4, 8, 12, 16, 20, ...

(4 × 1, 4 × 2, 4 × 3, ...)


LCM tells us the smallest common multiple.
2. LCM (Lowest Common Multiple)

LCM is the smallest number that is a multiple of two or more numbers.


Example:

Multiples of 12 = 12, 24, 36, 48, 60, ...

Multiples of 18 = 18, 36, 54, 72, 90, ...


Common multiples = 36, 72, 108, ...

Smallest of these is 36.

So, LCM of 12 and 18 = 36
HOW TO FIND HCF? (TWO EASY METHODS)
METHOD 1: LISTING FACTORS
  1. Step 1: Find the factors of each number.
  2. Step 2: Find the common factors.
  3. Step 3: Choose the greatest one.

Example: Find 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


Common factors = 1, 2, 3, 4, 6, 12

Greatest is 12.

HCF of 24 and 36 = 12
METHOD 2: PRIME FACTORISATION
  1. Step 1: Find the prime factorisation of each number.
  2. Step 2: Take the common prime factors.
  3. Step 3: Multiply them.

Example: Find the HCF of 24 and 36

24
2 × 12
2 × 6
2 × 3
36
2 × 18
2 × 9
3 × 3

Common prime factors = 2 and 3

HCF = 2 × 3 = 6

HCF of 24 and 36 = 6
HOW TO FIND LCM? (TWO EASY METHODS)
METHOD 1: LISTING MULTIPLES
  1. Step 1: List the multiples of each number.
  2. Step 2: Find the common multiples.
  3. Step 3: Choose the smallest one.

Example: Find the LCM of 6 and 8

Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, ...

Multiples of 8 = 8, 16, 24, 32, 40, 48, ...


Common multiples = 24, 48, 72, ...

Smallest is 24.

LCM of 6 and 8 = 24
METHOD 2: PRIME FACTORISATION
  1. Step 1: Find the prime factorisation of each number.
  2. Step 2: Take all the prime factors.
    (If a factor is common, take it the highest number of times.)
  3. Step 3: Multiply them.

Example: Find the LCM of 24 and 36

24
2 × 12
2 × 6
2 × 3
36
2 × 18
2 × 9
3 × 3
Take highest powers:
2 appears 3 times
3 appears 2 times

LCM = 2 × 2 × 2 × 3 × 3 = 72

LCM of 24 and 36 = 72
QUICK FACTS 💡
  • HCF is always less than or equal to the smaller number.
  • LCM is always greater than or equal to the greater number.
  • Every number is the HCF and LCM of itself.
  • HCF of two co-prime numbers (like 7 and 9) is 1.
  • LCM of two co-prime numbers is their product.
PRACTICE TIME! ✏️

A. Find the HCF of the following.

  1. 12 and 18
  2. 15 and 25
  3. 20 and 30
  4. 24 and 36
  5. 28 and 42

B. Find the LCM of the following.

  1. 4 and 6
  2. 5 and 10
  3. 6 and 9
  4. 8 and 12
  5. 9 and 15
REAL LIFE APPLICATIONS

1. HCF in real life:

If Riya has 24 pencils and Siya has 36 pencils. They want to make the biggest possible equal packs with no pencils left.

The number of pencils in each pack = HCF of 24 and 36 = 12

So, they can make 12 pencils in each pack.


2. LCM in real life:

A bell rings every 6 minutes.

A siren rings every 8 minutes.

After how many minutes will they ring together again?

LCM of 6 and 8 = 24

They will ring together after 24 minutes.

THINK AND SOLVE! 🤔

1. The HCF of two numbers is 9 and their LCM is 72. If one number is 18, what is the other number?


2. The LCM of two numbers is 60 and their HCF is 5. If one number is 20, what is the other number?