Playing with Numbers — Class 6 IMO

Factors, Multiples, Prime & Composite

A factor divides a number exactly; a multiple is the result of multiplying by 1, 2, 3, … A prime number has exactly two factors (1 and itself); a composite number has more than two. The number 1 is neither.

Example 1: List the factors of 24.

1, 2, 3, 4, 6, 8, 12, 24.

Example 2: Is 1 a prime number?

No — 1 has only one factor, so it is neither prime nor composite.

Quick recap

Factor divides exactly; multiple comes from multiplying.
Prime: exactly two factors; composite: more than two; 1 is neither.

✓ Quick check

Which of these is a prime number?

23 has only the factors 1 and 23, so it is prime.

Which number is a common factor of 12 and 18?

6 divides both 12 and 18 exactly.

Divisibility Rules & Prime Factorisation

What is Prime Factorization?

Prime factorization is breaking down a number into its prime factors – the prime numbers that multiply together to make the original number.

Prime Numbers: Numbers greater than 1 with exactly 2 factors (1 and itself)

Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29...

Composite Numbers: Numbers with more than 2 factors

Examples: 4, 6, 8, 9, 10, 12...

What is a Factor Tree?

A factor tree is a diagram that shows how a number breaks down into prime factors.

Steps to Build a Factor Tree:

Write the number at the top
Find two factors that multiply to make the number
Branch them down
Continue breaking composite numbers until all "leaves" are prime
Circle the prime numbers

FACTOR TREE FOR 24:

 24
 / 
 4 6
 / / 
 2 2 2 3
 
 Prime factors: 2, 2, 2, 3
 Written as: 2³ × 3


FACTOR TREE FOR 36:

 36
 / 
 6 6
 / / 
 2 3 2 3
 
 Prime factors: 2, 2, 3, 3
 Written as: 2² × 3²


FACTOR TREE FOR 48:

 48
 / 
 6 8
 / / 
 2 3 2 4
 / 
 2 2
 
 Prime factors: 2, 2, 2, 2, 3
 Written as: 2⁴ × 3


ANOTHER WAY TO BUILD (different factors):

 48
 / 
 4 12
 / / 
 2 2 3 4
 / 
 2 2
 
 Same result: 2⁴ × 3

Example 1: Is 348 divisible by 4?

The last two digits 48 ÷ 4 = 12, so yes.

Example 2: Write the prime factorisation of 36.

36 = 2 × 2 × 3 × 3.

Example 3:

Find the prime factorization of 30 using a factor tree.

``` 30 / 5 6 / 2 3

Prime factors: 5, 2, 3 Answer: 2 × 3 × 5 ```

Example 4:

Find the prime factorization of 72.

``` 72 / 8 9 / / 2 4 3 3 / 2 2

Prime factors: 2, 2, 2, 3, 3 Answer: 2³ × 3² ```

Example 5:

Find the prime factorization of 100.

``` 100 / 10 10 / / 2 5 2 5

Prime factors: 2, 2, 5, 5 Answer: 2² × 5² ```

Quick recap

Use divisibility rules for quick checks.
Prime factorisation = a number as a product of primes.
Prime numbers have exactly 2 factors
Factor tree breaks numbers down to primes
Order doesn't matter – multiplication is commutative
Use exponents for repeated factors
Every number has exactly one prime factorization

✓ Quick check

Is 1,287 divisible by 9?

Digit sum 1 + 2 + 8 + 7 = 18, which is divisible by 9, so yes.

What is the prime factorisation of 45?

45 = 3 × 3 × 5.

HCF & 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

Example 1: Find the HCF and LCM of 24 and 36.

HCF = 12 and LCM = 72.

Example 2: Find the HCF of 15 and 25.

Common factors 1 and 5, so HCF = 5.

Example 3:

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

Example 4:

Find LCM of 9 and 12 using prime factorization.

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

Example 5:

Find LCM of 5 and 7.

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

Example 6:

Find LCM(8, 12) using GCF formula.

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

Quick recap

HCF = largest common factor.
LCM = smallest common multiple.
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

What is the LCM of 4 and 5?

The smallest common multiple of 4 and 5 is 20.

What is the HCF of 18 and 24?

Common factors of 18 and 24 give HCF = 6.