Prime Factorization, LCM, GCF • Topic 2 of 3

Greatest Common Factor - GCF

What is GCF?

The Greatest Common Factor (GCF) is the largest number that divides evenly into two or more numbers.

Method 1: List Factors

Find GCF 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
Greatest common factor: 6

Method 2: Prime Factorization

Find GCF of 18 and 24:

18 = 2 × 3²
24 = 2³ × 3
Take the smallest power of each common prime:
For 2: min(1, 3) = 2¹
For 3: min(2, 1) = 3¹
GCF = 2 × 3 = 6

Method 3: Factor Tree Method

       18          24
      /  \        /  \
     2    9      4    6
         / \    / \  / \
        3   3  2  2 2  3
        
    18 = 2 × 3²
    24 = 2³ × 3
    GCF = 2 × 3 = 6

FINDING GCF - VENN DIAGRAM METHOD:

    Place prime factors in a Venn diagram
    
    GCF(18, 24):
    
          18                24
        (2 × 3²)          (2³ × 3)
        
              ┌─────┬─────┐
              │ 18  │ 24  │
              │     │     │
           ┌──┼───┐ │ ┌───┼──┐
           │  │   │ │ │   │  │
           │  │ 3 │ │ │ 2 │  │
           │  │   │ │ │   │  │
           │  └───┼─┼─┼───┘  │
           │      │ │ │      │
           │   2  │3│ 2²    │
           │      │ │ │      │
           └──────┴─┴─┴──────┘
           
    Intersection (common): 2 and 3
    GCF = 2 × 3 = 6


USING GCF TO SIMPLIFY FRACTIONS:

    18/24 = ?
    
    GCF(18,24) = 6
    
    18 ÷ 6 = 3
    24 ÷ 6 = 4
    
    18/24 = 3/4 ✓


REAL-LIFE APPLICATION:

    You have 18 red balloons and 24 blue balloons.
    You want to make identical bouquets with the same number of each color.
    What's the largest number of bouquets?
    
    Answer: GCF(18,24) = 6 bouquets
    Each bouquet: 3 red, 4 blue balloons

Solved Examples

Worked Example

Find the GCF of 12 and 18.

Solution (Listing factors):

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

Solution

Worked Example

Find the GCF of 24 and 36 using prime factorization.

Solution

24 = 2³ × 3
36 = 2² × 3²
Common primes: 2² and 3¹
GCF = 4 × 3 = 12

Worked Example

Find the GCF of 15 and 28.

Solution

15 = 3 × 5
28 = 2² × 7
No common prime factors!
GCF = 1 (they are relatively prime)

Worked Example

Find GCF(48, 60).

Solution

48 = 2⁴ × 3
60 = 2² × 3 × 5
Common: 2² (4) and 3
GCF = 4 × 3 = 12

Key Points

GCF = largest number that divides all given numbers
Method 1: List all factors
Method 2: Prime factorization (take smallest powers)
Method 3: Factor tree + compare
If no common factors, GCF = 1
GCF helps simplify fractions and solve grouping problems

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The GCF is the ___ factor common to two numbers.

Explanation: Largest common factor.

Q2.The GCF of $12$ and $18$ is:

Explanation: $6$.

Q3.The GCF of two different prime numbers is:

Explanation: $1$.

Q4.The GCF of $8$ and $12$ is:

Explanation: $4$.

Practice more MCQs → 📝 Assignment · 35 marks