Example 1
Prime-factorise 360
- Divide 360 by 10 to write 360 = 36 × 10.
- Factor 36 = 2² × 3².
- Factor 10 = 2 × 5.
- Combine equal primes: 360 = 2³ × 3² × 5.
Final answer: 360 = 2³ × 3² × 5.
Check: Evaluate 8 × 9 × 5 = 360.
What students learn in AC9M7N02
We are learning to represent natural numbers as products of powers of prime numbers using exponent notation.
A prime number has exactly two positive factors: one and itself. Prime factorisation expresses a natural number as a product of primes and uses exponent notation to compress repeated factors.
Different factor-tree paths for the same number finish with the same prime factors. For example, 360 = 2³ × 3² × 5, regardless of whether the first split is 36 × 10 or another valid pair.
Prime powers reveal divisibility and support highest common factor and least common multiple reasoning. An HCF selects shared primes with smaller exponents; an LCM selects every required prime with its greatest exponent.
Success criteria
- I can decompose a natural number into prime factors.
- I can write repeated prime factors using exponent notation.
- I can use prime powers to justify divisibility, HCF and LCM.
Key vocabulary
- prime number
- A natural number greater than one with exactly two positive factors: one and itself.
- prime factorisation
- A natural number written as a product containing only prime factors.
- exponent
- The small raised number showing how many times a factor is used in repeated multiplication.
- power
- An expression made from a base and exponent to represent repeated multiplication.
- highest common factor (HCF)
- The greatest natural number that divides each given number exactly.
- least common multiple (LCM)
- The smallest positive multiple shared by all the given numbers.
- factor tree
- A branching diagram that repeatedly splits a composite number until every leaf is prime.
Visual models and representations
Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.
4 worked numerical & application examples
Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.
Example 2
Use prime powers to find an LCM
- Write 45 = 3² × 5.
- Write 60 = 2² × 3 × 5.
- Select the greatest exponent of each prime: 2², 3² and 5.
- Multiply 4 × 9 × 5 = 180.
Final answer: LCM(45,60) = 180.
Check: Both 45 and 60 divide 180 exactly.
Example 3
Application problem 1
Problem: Two flashing lights repeat every 72 s and 90 s. Use prime factorisation to find when they next flash together, showing the prime powers you use.
- Plan: Represent the information first, then calculate, interpret and independently check the result.
- Work: 72 = 2³×3² and 90 = 2×3²×5. The LCM is 2³×3²×5 = 360, so they flash together after 360 s.
- Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Final answer: 72 = 2³×3² and 90 = 2×3²×5. The LCM is 2³×3²×5 = 360, so they flash together after 360 s.
Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Example 4
Application problem 2
Problem: A school packs 72 red and 108 blue counters into the greatest possible number of identical kits with no leftovers. Find the number of kits and each kit's contents.
- Plan: Prime-factorise both totals, find the HCF, then divide each total by it.
- Work: Since 72 = 2³×3² and 108 = 2²×3³, the HCF is 2²×3² = 36; each of 36 kits has 2 red and 3 blue counters.
- Interpret: The highest common factor gives the greatest number of equal kits.
Final answer: Since 72 = 2³×3² and 108 = 2²×3³, the HCF is 2²×3² = 36; each of 36 kits has 2 red and 3 blue counters.
Check: The highest common factor gives the greatest number of equal kits.
Common misconceptions
Common mistake: One treated as prime.
Correction: 1 has one positive factor, not two.
Common mistake: Composite leaves remain in a factor tree.
Correction: Continue until every leaf is prime.
Common mistake: Exponents added within one factorisation.
Correction: The exponent counts repeated identical prime factors.
10 important problems to solve
Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.
1. Prime-factorise 24 using exponent notation.
Check answer
Answer: 24 = 2³ × 3.
Hint: Divide by the smallest prime repeatedly.
Why: Repeated prime factors are compressed with exponents.
2. Prime-factorise 36.
Check answer
Answer: 36 = 2² × 3².
Hint: Use 36 = 4 × 9.
Why: Every final factor is prime.
3. Explain why 1 is not prime.
Check answer
Answer: One has only one positive factor, but a prime has exactly two.
Hint: List the positive factors of 1.
Why: Prime classification depends on factor count.
4. Prime-factorise 84.
Check answer
Answer: 84 = 2² × 3 × 7.
Hint: Start with 84 ÷ 2.
Why: The product of prime powers reconstructs 84.
5. Find the HCF of 48 and 72 using prime factors.
Check answer
Answer: 48 = 2⁴×3 and 72 = 2³×3², so HCF = 2³×3 = 24.
Hint: Compare exponents prime by prime.
Why: HCF uses shared primes with smaller exponents.
6. Find the LCM of 45 and 60.
Check answer
Answer: LCM = 2²×3²×5 = 180.
Hint: Prime-factorise both numbers first.
Why: LCM uses every prime at its greatest exponent.
7. Is 540 divisible by 18? Justify with prime powers.
Check answer
Answer: Yes: 540 = 2²×3³×5 contains 18 = 2×3².
Hint: Compare the exponent of each prime.
Why: A number is divisible when it contains all required prime factors.
8. Find the smallest number divisible by 12, 18 and 30.
Check answer
Answer: The prime forms are 12=2²×3, 18=2×3² and 30=2×3×5, so the LCM is 2²×3²×5=180.
Hint: Factor all three numbers.
Why: Greatest prime exponents produce the least common multiple.
9. A school packs 72 red and 108 blue counters into the greatest possible number of identical kits with no leftovers. Find the number of kits and each kit's contents.
Check answer
Answer: Since 72 = 2³×3² and 108 = 2²×3³, the HCF is 2²×3² = 36; each of 36 kits has 2 red and 3 blue counters.
Hint: Prime-factorise both totals, find the HCF, then divide each total by it.
Why: The highest common factor gives the greatest number of equal kits.
10. Two flashing lights repeat every 72 s and 90 s. Use prime factorisation to find when they next flash together, showing the prime powers you use.
Check answer
Answer: 72 = 2³×3² and 90 = 2×3²×5. The LCM is 2³×3²×5 = 360, so they flash together after 360 s.
Hint: Represent the information first, then calculate, interpret and independently check the result.
Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.
Curriculum coverage and elaborations
Content description: represent natural numbers as products of powers of prime numbers using exponent notation.
International curriculum mapping
The Australian Curriculum code above is exact. Victorian Year 7, NSW Stage 4, US Grade 7, England Key Stage 3, New Zealand Level 4 and comparable international curricula contain broadly related learning, but code-to-code equivalence varies by jurisdiction.
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Prime Factorization
Math Antics — Break a natural number into prime factors before recording repeated factors with powers.
As you watch: Why do factor trees end with prime numbers?
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Try it: Express 72 as a product of prime factors, then write repeated factors using exponent notation.
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Represent natural numbers as products of powers of prime numbers...
Mapped skill: represent natural numbers as products of powers of prime numbers using exponent notation
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M7N02 · Year 7 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M7N02 · Level 7 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA4-IND-C-01 · Stage 4 |
| United States (USA) | Common Core State Standards for Mathematics | Grade 7 |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 7 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 8, Key Stage 3 |
| India | NCERT / CBSE — Mathematics | Class 7 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M7N02 — Prime Factorisation with Exponent Notation
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