Lesson: HCF and Factors

1. What Are Factors?

A factor of a number is a number that divides it exactly (no remainder).

Examples:

• Factors of 12 → 1, 2, 3, 4, 6, 12
• Factors of 30 → 1, 2, 3, 5, 6, 10, 15, 30

How to Find Factors

1. Start from 1
2. Check which numbers divide exactly
3. Go up to the square root
4. Write factor pairs

2. What Is HCF?

The Highest Common Factor (HCF) is the largest number that is a factor of two or more numbers.

Example:

Find the HCF of 18 and 24

• Factors of 18 → 1, 2, 3, 6, 9, 18
• Factors of 24 → 1, 2, 3, 4, 6, 8, 12, 24
• Common factors → 1, 2, 3, 6

HCF = 6

3. Using the Prime Factor Method

Example:

Find the HCF of 36 and 60

• 36 = 2 × 2 × 3 × 3
• 60 = 2 × 2 × 3 × 5
• Common primes: 2, 2, 3

HCF = 2 × 2 × 3 = 12

Practice Questions

A. Find the factors

1. List all factors of 28
2. List all factors of 45
3. List all factors of 32

B. Find the HCF

4. HCF of 20 and 30
5. HCF of 15 and 35
6. HCF of 48 and 64

C. Using Prime Factorization

7. HCF of 42 and 70
8. HCF of 54 and 81
9. HCF of 24, 36, and 60

D. Word Problem

10. A teacher has 36 red pencils and 48 blue pencils.
    She wants to make identical sets using all pencils. What is the
    maximum number of sets she can make?
    
    (Hint: Find HCF of 36 and 48)

Answer Key

A. Factors

1. Factors of 28 → 1, 2, 4, 7, 14, 28
2. Factors of 45 → 1, 3, 5, 9, 15, 45
3. Factors of 32 → 1, 2, 4, 8, 16, 32

B. HCF

4. HCF(20, 30) = 10
5. HCF(15, 35) = 5
6. HCF(48, 64) = 16

C. Prime Factorization

7. HCF(42, 70) = 14
   (42 = 2 × 3 × 7, 70 = 2 × 5 × 7 → Common: 2, 7)
8. HCF(54, 81) = 27
   (54 = 2 × 3³, 81 = 3⁴ → Common: 3³ = 27)
9. HCF(24, 36, 60) = 12
   (Common prime factors: 2² × 3 = 12)

D. Word Problem

10. HCF(36, 48) = 12
    So the teacher can make 12 identical pencil sets.