IT'S PRIME TIME!
PRIME FACTORISATION WORKBOOK
Welcome, Math Explorer!
Master prime factorisation with this fantastic resource, designed to engage students of all levels! Packed with easy-to-understand explanations, guided examples and a whole heap of practice questions, this ebook will help you to conquer prime numbers, factors, and prime factorisation.
Clear Explanations
Simplified steps to break down complex concepts
Vibrant Design
Eye-catching visuals make learning fun and memorable
Hands-On Practice
Tons of examples and practice questions to solidify learning
Multi-sensory Learning
Visual and interactive elements for better understanding
Who It's For
- Students of all ages looking to conquer an important maths topic that features in SATs, 11-plus and GCSEs and everyday life!
- Homeschoolers looking for a fun and structured resource to make maths more engaging
- Teachers looking for a ready-to-use classroom resource with examples, questions and visual aids
- Parents hoping to brush up their skills to support their child's learning
Chapter 1: The Building Blocks
What are Prime Numbers?
A Prime Number is a number greater than 1 that has only two factors: 1 and itself.
Examples:
- 5 is prime because its only factors are 1 and 5.
- 7 is prime because its only factors are 1 and 7.
A Composite Number has more than two factors.
- 6 is composite because its factors are 1, 2, 3, and 6.
Prime numbers (red) vs Composite numbers (green)
What are Factors?
Factors are numbers you multiply together to get another number.
Example: The factors of 12 are 1, 2, 3, 4, 6, and 12.
Because: 1 × 12 = 12, 2 × 6 = 12, and 3 × 4 = 12.
So, 12 = 2 × 2 × 3 = 2² × 3
Chapter 2: Divisibility Rules - Your Secret Shortcuts!
Divisible by 2
The number is even (ends in 0, 2, 4, 6, 8)
Example: 58 ends in 8, so YES
Divisible by 3
The sum of the digits is divisible by 3
Example: 123: 1+2+3=6, 6÷3=2, so YES
Divisible by 5
The number ends in 0 or 5
Example: 95 ends in 5, so YES
Divisible by 9
The sum of the digits is divisible by 9
Example: 234: 2+3+4=9, 9÷9=1, so YES
Your Turn! Circle the numbers that are divisible.
Divisible by 3?
84, 45, 105, 72, 112, 41, 34
Hint: Add the digits and check if the sum is divisible by 3
Divisible by 9?
57, 72, 88, 96, 108, 117
Hint: Add the digits and check if the sum is divisible by 9
Divisible by 5?
45, 100, 55
Hint: Check if the number ends in 0 or 5
Chapter 3: The Main Event - Prime Factorisation
Prime Factorisation is like breaking a number down into its smallest possible building blocks – the prime numbers that multiply together to make the original number.
Every composite number can be written as a product of its prime factors.
Method 1: The Factor Tree
This is a visual way to break the number down step-by-step.
Let's factorise 60 using a factor tree:
- Find any two factors of 60 (except 1 and 60). We used 6 and 10.
- Break down 6 into 2 and 3 (both prime!).
- Break down 10 into 2 and 5 (both prime!).
- The prime factors are all the prime numbers at the ends of the branches.
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Method 2: The Division Ladder (Repeated Division)
This is a neat and tidy method using division.
Let's factorise 60 using the division ladder:
- Divide 60 by the smallest prime number that goes into it (2). Write the result (30) below.
- Divide 30 by the smallest prime that goes into it (2 again). Write the result (15) below.
- 15 is not divisible by 2, so move to the next smallest prime (3). 15 ÷ 3 = 5. Write 5 below.
- 5 is a prime number. Divide it by itself. 5 ÷ 5 = 1. Stop when you reach 1.
- The prime factors are the divisors you used (the numbers on the left).
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5
Prime Factorisation Examples
| Number | Prime Factorisation | Step-by-Step Explanation |
|---|---|---|
| 6 | 2 × 3 | 6 ÷ 2 = 3; 3 ÷ 3 = 1. Primes: 2, 3. |
| 8 | 2³ | 8 ÷ 2 = 4; 4 ÷ 2 = 2; 2 ÷ 2 = 1. Primes: 2, 2, 2. |
| 12 | 2² × 3 | 12 ÷ 2 = 6; 6 ÷ 2 = 3; 3 ÷ 3 = 1. Primes: 2, 2, 3. |
| 19 | 19 | Prime, so itself. |
| 32 | 2⁵ | 32 ÷ 2 = 16; 16 ÷ 2 = 8; 8 ÷ 2 = 4; 4 ÷ 2 = 2; 2 ÷ 2 = 1. |
Chapter 4: Practice Makes Perfect!
Part A: Find the Prime Factorisation. Use either method!
1. 36
2. 50
3. 81
4. 100
5. 66
Part B: Challenge Questions!
6. Which number has the prime factorisation 2² × 3²?
7. Two numbers have prime factorisations 2 × 3 × 5 and 2 × 5 × 7. What is the largest number that divides evenly into both (the Greatest Common Divisor or GCD)?
Answer Key
Chapter 2: Divisibility Rules
- Divisible by 3: 84, 45, 105, 72
- Divisible by 9: 72, 108, 117
- Divisible by 5: 45, 100, 55 (All of them!)
Chapter 4: Practice Makes Perfect
- 36 = 2 × 2 × 3 × 3 = 2² × 3²
- 50 = 2 × 5 × 5 = 2 × 5²
- 81 = 3 × 3 × 3 × 3 = 3⁴
- 100 = 2 × 2 × 5 × 5 = 2² × 5²
- 66 = 2 × 3 × 11 = 2 × 3 × 11
Challenge Questions:
- 2² × 3² = 4 × 9 = 36
- The common prime factors are 2 and 5. So the GCD is 2 × 5 = 10.
Certificate of Completion
This certifies that
_________________________
has successfully completed the
Prime Factorisation Workbook
and is now a certified
Prime Number Pro!
Date: _________________________
Instructor: M.RajaRao, MSc, MEd