Prime Factors & Exponent Notation
Prime or Composite?
Sort each number into the correct column.
Match to Prime Factorisation
Draw a line from each number to its prime factorisation.
Identify the Prime Factorisation
Circle the correct prime factorisation for each number.
24
36
50
72
Exponent Notation Values
Draw a line to match each exponent expression to its value.
Expanded Form
Circle the correct expanded form for each expression.
2⁴
3³
5²
2⁵
Group by Number of Prime Factors
Sort each number by how many prime factors it has (counting repeats).
Powers of 2 and 3
Fill in the missing values in each sequence of powers.
Show Your Factor Tree
Draw a factor tree and write the prime factorisation using exponents for each number.
Factor tree for 48:
Prime factorisation of 48:
Factor tree for 100:
Prime factorisation of 100:
Exponent Notation Shorthand
Match each expanded product to its exponent notation form.
Highest Common Factor
Use prime factorisation to find the HCF. Circle the correct answer.
HCF of 12 and 18
HCF of 20 and 30
HCF of 24 and 36
HCF of 15 and 35
LCM Using Prime Factors
Draw a line to match each pair of numbers to their lowest common multiple.
Explain Your Thinking
Answer each question in your own words.
Why is 1 not considered a prime number?
A number has prime factorisation 2³ × 7. What is the number?
Write a number between 50 and 60 and find its prime factorisation.
Compare Exponent Expressions
Circle the correct comparison symbol (<, > or =).
2⁵ ○ 5²
3³ ○ 4²
2⁸ ○ 10²
6² ○ 2⁶
Order by Value
Sort these expressions from smallest to largest value.
Factor Tree — More Numbers
Build factor trees and write prime factorisations for each number.
Factor tree for 90:
Prime factorisation of 90:
Factor tree for 126:
Prime factorisation of 126:
HCF Matching
Match each pair to its HCF.
Prime Factorisation Word Problems
Use prime factorisation to solve each problem.
Buses on Route A leave every 12 minutes. Buses on Route B leave every 8 minutes. They both leave at 9:00 am. Using LCM, when is the next time they leave together?
A school hall can be arranged in rows of 24 or rows of 36 with no seats left over. What is the smallest number of seats the hall could have?
Product of Primes
Circle the expression that correctly represents the number as a product of primes.
84
180
210
Classify by Prime Factorisation Complexity
Sort each number by how many distinct prime factors it has.
Powers of 5 and 10
Fill in the missing terms.
HCF in Real Life
Solve each problem using HCF.
You have 48 apples and 36 oranges. You want to make identical fruit bags with no fruit left over. What is the greatest number of bags you can make? How many of each fruit goes in each bag?
A rectangular room is 48 cm × 60 cm (in a model). What is the largest square tile that fits perfectly with no cutting? (Use HCF.)
Powers and Their Roots
Match each power expression to the operation that undoes it.
LCM Word Problems
Write a calculation and solve.
A traffic light changes red every 30 seconds and green every 45 seconds. They both change at the same time at noon. When is the next time they change simultaneously?
Three runners complete a lap in 4, 6, and 9 minutes respectively. If they all start at the same time, how long until they are all at the start line together again?
Prime Factorisation of 3-digit Numbers
Circle the correct prime factorisation.
120
144
210
250
Three-Number HCF and LCM
Find the HCF and LCM of three numbers using prime factorisation.
Find HCF(12, 18, 30). Show all prime factorisations.
Find LCM(4, 6, 10). Show all prime factorisations.
Exponent Laws Investigation
Test the law aᵐ × aⁿ = aᵐ⁺ⁿ with specific values.
Calculate 2³ × 2⁴ by expanding each, then by adding exponents. Do both methods give the same answer?
Does (2 × 3)² = 2² × 3²? Show both sides.
Classify: Perfect Squares, Perfect Cubes, or Neither
Sort each number.
Zero and Negative Exponents Preview
Circle the correct value.
5⁰ =
2⁰ =
3⁰ =
10⁰ =
Factor Tree Reverse — Work Backwards
Given the prime factorisation, write the number and list all its factors.
2³ × 3 = ? List all factors of this number.
2² × 5 × 7 = ? List all factors of this number.
Match: Expression to Simplified Form
Simplify each expression using the laws of exponents. Match.
Cryptography Connection
Explore how large prime numbers are used in internet security.
Find the prime factorisation of 1001. Show your working.
RSA encryption uses the product of two large primes. If p = 11 and q = 13, what is p × q? Why is it hard to reverse this process for very large primes?
Abundant, Perfect and Deficient Numbers
The sum of proper divisors of a number (all factors except the number itself) can be less than, equal to, or greater than the number.
Find all proper divisors of 6. Is 6 perfect, abundant, or deficient?
Find all proper divisors of 12. Is 12 perfect, abundant, or deficient?
Find all proper divisors of 28. Is 28 perfect, abundant, or deficient?
Goldbach's Conjecture Investigation
Goldbach's Conjecture states that every even integer greater than 2 can be written as the sum of two primes.
Test the conjecture for: 10, 16, 28, 40. Write each as a sum of two primes.
Is the conjecture proven? Look it up and write a short explanation.
Prime Factorisation of Factorials
Factorial notation: n! = n × (n−1) × … × 2 × 1.
Find the prime factorisation of 5! = 120.
Find the prime factorisation of 6! = 720.
How many times does the prime 2 appear in the factorisation of 8! ? Explain your method.
Mersenne Primes
Mersenne primes have the form 2ⁿ − 1. Sort each expression as Mersenne prime or not.
Create Your Own HCF/LCM Problem
Design a real-world word problem involving HCF or LCM.
Write a word problem whose solution requires finding the LCM of two numbers. Include the solution.
Write a word problem whose solution requires finding the HCF of two numbers. Include the solution.
Project: My Own Sieve of Eratosthenes
Recreate the Sieve of Eratosthenes to find all primes up to 100.
Write the numbers 1–100 in a 10×10 grid. Cross out 1. Circle 2 and cross out all its multiples. Circle 3 and cross out all its multiples. Continue for 5 and 7. List all circled (prime) numbers below.
How many primes did you find between 1 and 100?
Why is it enough to test primes up to √100 = 10 to find all primes below 100?
Prime Factorisation at Home
Look for numbers around you and practise prime factorisation.
- 1Pick any page number from a book and write its full prime factorisation.
- 2Find a number on a product label (e.g., weight in grams) and build its factor tree.
- 3Challenge a family member: give them a prime factorisation and have them guess the number.
- 4List all factors of 60 using its prime factorisation 2² × 3 × 5. How many factors does it have?
- 5Use the GIMPS website (mersenne.org) to read about the search for the largest prime.