Year 7 Mathematics · AC9M7N02

Prime Factorisation with Exponent Notation

We are learning to represent natural numbers as products of powers of prime numbers using exponent notation

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Learning goalsSay it simply

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 conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Prime-factorise 360

  1. Divide 360 by 10 to write 360 = 36 × 10.
  2. Factor 36 = 2² × 3².
  3. Factor 10 = 2 × 5.
  4. Combine equal primes: 360 = 2³ × 3² × 5.

Final answer: 360 = 2³ × 3² × 5.

Check: Evaluate 8 × 9 × 5 = 360.

Example 2

Use prime powers to find an LCM

  1. Write 45 = 3² × 5.
  2. Write 60 = 2² × 3 × 5.
  3. Select the greatest exponent of each prime: 2², 3² and 5.
  4. 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.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: 72 = 2³×3² and 90 = 2×3²×5. The LCM is 2³×3²×5 = 360, so they flash together after 360 s.
  3. 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.

  1. Plan: Prime-factorise both totals, find the HCF, then divide each total by it.
  2. 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.
  3. 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.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7N02 - Prime Factorisation with Exponent Notation
Example 1

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.

Example 2

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

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

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.

Curriculum examplesCopied content

Content description: represent natural numbers as products of powers of prime numbers using exponent notation.

Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 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. 2. Prime-factorise 36.

    Check answer

    Answer: 36 = 2² × 3².

    Hint: Use 36 = 4 × 9.

    Why: Every final factor is prime.

  3. 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. 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. 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. 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. 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. 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. 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. 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.

Practice and reviewReady for practice

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.

Curriculum alignmentStart here

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.
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