Prime Factors, HCF and LCM Revision Notes for OCR GCSE Maths

Prime Factors, HCF and LCM

1. Prime Factors

Definition: A prime factor is a factor of a number that is a prime number. Any positive integer can be expressed as a product of its prime factors. This process is called prime factorization.

Factor Trees:

A factor tree is a diagram that helps you break down a number into its prime factors by dividing the number into pairs of factors, and then continuing to divide any composite numbers until all the factors are prime.

Important Note:

$1$ is NOT a prime number, so it will never appear in your factor tree.

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Worked Example 1: Question:

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

Solution:

Start with $60$ :

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

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: $60→(2×3)×(2×5)$

Stop when you reach prime numbers:

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

Write the final product of prime factors:

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

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.

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Worked Example 2: Question:

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

Solution:

Start with $360$ :

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

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: $360=2³×3²×5$

Check Your Answer:

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

2. Highest Common Factor (HCF)

Definition: The HCF of two numbers is the largest number that divides exactly into both of them without leaving a remainder.

3. Lowest Common Multiple (LCM)

Definition: The LCM of two numbers is the smallest number that is a multiple of both numbers.

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

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:

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

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

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

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.

\begin{array}{c|c}\text{Circle for 24} & \text{Circle for 40} \\\hline3 & 5\end{array}

The overlapping section:

                                                                 $2³=8$

Finding the HCF

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

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 $40:$ $:highlight[120]$

Prime Factors, HCF and LCM (OCR GCSE Maths): Revision Notes