Prime Factorization, LCM, GCF • Topic 3 of 3

Least Common Multiple - LCM

What is LCM?

The Least Common Multiple (LCM) is the smallest positive number that is a multiple of two or more numbers.

Method 1: List Multiples

Find LCM of 4 and 6:

Multiples of 4: 4, 8, 12, 16, 20, 24...
Multiples of 6: 6, 12, 18, 24, 30...
Common multiples: 12, 24, 36...
Least common multiple: 12

Method 2: Prime Factorization

Find LCM of 4 and 6:

4 = 2²
6 = 2 × 3
Take the largest power of each prime:
For 2: max(2, 1) = 2² = 4
For 3: max(0, 1) = 3¹ = 3
LCM = 4 × 3 = 12

Method 3: Using GCF Formula

LCM(a, b) = (a × b) ÷ GCF(a, b)

Example: LCM(4,6) = (4×6) ÷ GCF(4,6)
                = 24 ÷ 2 = 12

MULTIPLES ON NUMBER LINE - LCM(4,6):

    Multiples of 4:    0    4    8    12   16   20   24
                      │    │    │    │    │    │    │
    Multiples of 6:    0    6    12   18   24
                      │    │    │    │    │
                      
    First common: 12 ← LCM


PRIME FACTORIZATION METHOD - LCM(12, 18):

    12 = 2² × 3
    18 = 2 × 3²
    
    Take largest powers:
    2² = 4
    3² = 9
    
    LCM = 4 × 9 = 36
    
    Check: 36 ÷ 12 = 3, 36 ÷ 18 = 2 ✓


USING VENN DIAGRAM FOR LCM:

    Place prime factors in a Venn diagram
    
          12                18
        (2² × 3)          (2 × 3²)
        
              ┌─────┬─────┐
              │ 12  │ 18  │
              │     │     │
           ┌──┼───┐ │ ┌───┼──┐
           │  │   │ │ │   │  │
           │  │ 3 │ │ │ 2 │  │
           │  │   │ │ │   │  │
           │  └───┼─┼─┼───┘  │
           │      │ │ │      │
           │   2  │3│ 2²    │
           │      │ │ │      │
           └──────┴─┴─┴──────┘
           
    LCM = multiply all primes in the diagram
    = 2² × 3² = 4 × 9 = 36


REAL-LIFE APPLICATION:

    Bus A comes every 4 minutes.
    Bus B comes every 6 minutes.
    They both leave at the same time.
    When will they next leave together?
    
    Answer: LCM(4,6) = 12 minutes

Solved Examples

Worked Example

Find the LCM of 6 and 8.

Solution (Listing multiples):

Multiples of 6: 6, 12, 18, 24, 30...
Multiples of 8: 8, 16, 24, 32...
First common: 24

Solution

Worked Example

Find LCM of 9 and 12 using prime factorization.

Solution

9 = 3²
12 = 2² × 3
Largest powers: 2² (4), 3² (9)
LCM = 4 × 9 = 36

Worked Example

Find LCM of 5 and 7.

Solution

5 and 7 are both prime and different
They share no common factors
LCM = 5 × 7 = 35

Worked Example

Find LCM(8, 12) using GCF formula.

Solution

GCF(8,12) = 4
LCM = (8 × 12) ÷ 4 = 96 ÷ 4 = 24

Key Points

LCM = smallest number that is a multiple of all given numbers
Method 1: List multiples until common found
Method 2: Prime factorization (take largest powers)
Method 3: (a×b) ÷ GCF(a,b)
LCM helps find when events repeat/synchronize
For primes: LCM = product

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The LCM is the ___ multiple common to two numbers.

Explanation: Smallest common multiple.

Q2.The LCM of $4$ and $6$ is:

Explanation: $12$.

Q3.The LCM of two co-prime numbers is:

Explanation: Their product.

Q4.The LCM of $3$ and $5$ is:

Explanation: $15$.

Practice more MCQs → 📝 Assignment · 35 marks