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Stage 4 · Indices · MA4-IND-C-01

Primes, Divisibility and Factorisation

Think first: Could two different factor trees give different prime building blocks for 360?

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Know

  • prime and composite numbers
  • divisibility tests for 2, 3, 4, 5, 6 and 10
  • prime-factor index notation

Understand

  • why every whole number greater than 1 has a unique prime factorisation
  • why different valid factor trees finish with the same prime factors

Can do

  • test divisibility efficiently
  • build and check a factor tree
  • write a number as a product of prime powers
1

Divisibility before division

A prime number has exactly two positive factors: 1 and itself. A composite number has more than two. The number 1 is neither prime nor composite.

2 last digit even
3 digit sum divisible by 3
4 last two digits divisible by 4
5 last digit 0 or 5
6 divisible by 2 and 3
10 last digit 0
2

Prime factorisation

Split a composite number into factors, then keep splitting until every branch is prime. For 360:

360 = 36 × 10 = (2² × 3²) × (2 × 5) = 2³ × 3² × 5

Check by evaluating: 8 × 9 × 5 = 360. The superscripts count repeated prime factors.

3

Using the factorisation

For 72 = 2³ × 3² and 120 = 2³ × 3 × 5, the HCF uses common primes with the smaller indices: 2³ × 3 = 24. The LCM uses every prime with the larger indices: 2³ × 3² × 5 = 360.

Worked examples

Worked example 1: Test 4,356 for divisibility by 6

It is even, and 4 + 3 + 5 + 6 = 18 is divisible by 3. Therefore 4,356 is divisible by 6.

Worked example 2: Prime-factorise 540

540 = 54 × 10 = (2 × 3³) × (2 × 5) = 2² × 3³ × 5. Check: 4 × 27 × 5 = 540.

Worked example 3: Find HCF(84, 126)

84 = 2² × 3 × 7 and 126 = 2 × 3² × 7. Common minimum powers give 2 × 3 × 7 = 42.

Check your understanding

1. Which number is prime?

2. Which number is divisible by 6?

3. What is the prime factorisation of 360?

4. What is HCF(18, 24)?

5. Two correct factor trees start differently. What must be true?

Reason and apply

1. Use divisibility tests to decide whether 7,452 is divisible by 2, 3, 4, 5, 6 and 10.

Model response: It is divisible by 2 (even), 3 (digit sum 18), 4 (52 divisible by 4) and 6 (both 2 and 3). It is not divisible by 5 or 10.

2. Write 756 as a product of prime powers.

Model response: 756 = 2² × 3³ × 7. Check: 4 × 27 × 7 = 756.

3. Find the HCF and LCM of 48 and 180 using prime factors.

Model response: 48 = 2⁴ × 3; 180 = 2² × 3² × 5. HCF = 2² × 3 = 12. LCM = 2⁴ × 3² × 5 = 720.

4. Explain why 1 is not prime.

Model response: A prime must have exactly two positive factors. The number 1 has only one positive factor, so it is neither prime nor composite.