Worked Example 1: Question:

Express $60$ as a product of its prime factors.

Solution:

1. Start with $60$:

• Choose any two factors of $60$ to start the factor tree. You could start with $6×10$ or $12×5.$

2. Create the Factor Tree:

• For $6×10:$ $60→6×10$

• Now, break down $6$ into its prime factors: $6→2×3$

• Next, break down $10$ into its prime factors: $10→2×5$

• The factor tree now shows: $60→(2×3)×(2×5)$

3. Stop when you reach prime numbers:

• The factor tree is complete when all the branches end in prime numbers.

4. Write the final product of prime factors:

• The prime factorization of $60$ is: $60=2×2×3×5$
• Or, using Indices: $60=2²×3×5$

5. Check Your Answer:

• Multiply the prime factors back together to ensure they give the original number: $2×2×3×5=4×3×5=12×5=60$
• The answer is correct.

Worked Example 2: Question:

Express $360$ as a product of its prime factors.

Solution:

6. Start with $360$ :

• Begin by splitting $360$ into any two factors, such as $36×10$.

7. Create the Factor Tree:

• For $36×10$: $360→36×10$

• Break down $36$ and $10$ further: $36→6×6$ $10→2×5$

• Break down $6$: $6→2×3$

• The complete factor tree shows: $360=2×2×2×3×3×5$

• Alternatively, using Indices: $360=2³×3²×5$

8. Check Your Answer:

• Multiply the prime factors to verify: $2×2×2×3×3×5=8×9×5=72×5=360$

Worked Example Let's find the HCF and LCM of $24$ and $40$.

9. Prime Factorisation

• First, break down each number into its prime factors using a factor tree. For $24$:

                                         $24=2×2×2×3=2³×3$
  

For $40$:

                                          $40=2×2×2×5=2³×5$

Visual Representation:

10. Writing the Prime Factorisation Write the prime factorisations in a line:

                                       $For$ $24$: $24=2³×3$
    
                                       $For$ $40$: $40=2³×5$
    

11. Venn Diagram Method

• Draw two interlocking circles. Label one circle $24$ and the other $40$.
• Place the common factors (the ones that appear in both prime factorizations) in the middle section where the circles overlap.
• Place the remaining factors in the appropriate section of the circle.

The overlapping section:

                                                                 $2³=8$

12. Finding the HCF

• Multiply the numbers in the overlapping section: $HCF=2×2×2=8$

13. Finding the LCM

• Multiply all the numbers in the diagram: $LCM=2×2×2×3×5=120$

Conclusion

• HCF of $24$ and $40:$ $:highlight[8]$
• LCM of $24$ and $40:$ $:highlight[120]$