🦸 Factor Hero: Primes, Factors & Powers 🔢

Factors, multiples, primes, squares and cubes are the building blocks of number theory. Understanding these will help you simplify fractions, solve problems efficiently and spot patterns in mathematics. Every number has a story — the Factor Hero can tell it!

🌳 Prime Factorisation
36 = 2 × 2 × 3 × 3 = 2² × 3² (factor tree method)
🔢 HCF & LCM
HCF(12,18)=6 · LCM(4,6)=12 — use prime factors to find both!
⬛ Powers
5² = 25 (square) · 5³ = 125 (cube) · √25 = 5 · ∛125 = 5

📋 Golden Rules

1. Factors and Multiples

A factor of a number divides into it exactly (no remainder). A multiple is what you get when you multiply a number by an integer.

Factors of 12: 1, 2, 3, 4, 6, 12
Multiples of 12: 12, 24, 36, 48…

2. Prime Numbers

A prime number has exactly 2 factors: 1 and itself. The first primes are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29…
⚠️ Note: 1 is NOT a prime number (it only has 1 factor).

3. Prime Factorisation

Write any number as a product of prime numbers only. Use a factor tree — keep splitting until all branches are prime.

Example: 36 → 4 × 9 → (2 × 2) × (3 × 3) = 2² × 3²

4. HCF – Highest Common Factor

The HCF is the largest number that divides exactly into both numbers. Find it by listing factors or using prime factorisation.

HCF(12, 18): 12 = 2² × 3  |  18 = 2 × 3² → HCF = 2 × 3 = 6

5. LCM – Lowest Common Multiple

The LCM is the smallest number that both numbers divide into. Find it by listing multiples or using prime factorisation.

LCM(4, 6): multiples of 4: 4, 8, 12… multiples of 6: 6, 12… → LCM = 12

6. Square and Cube Numbers

Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100. Square roots reverse this.

Cube numbers: 1, 8, 27, 64, 125. Cube roots reverse this.

6² = 36  →  √36 = 6
4³ = 64  →  ∛64 = 4

7. Divisibility Rules

÷ 2: number is even (ends in 0, 2, 4, 6, 8)
÷ 3: digit sum is divisible by 3
÷ 4: last 2 digits form a number divisible by 4
÷ 5: ends in 0 or 5
÷ 9: digit sum is divisible by 9
÷ 10: ends in 0

Collins Stage 6 Cambridge Primary Maths — Unit 4: Multiples, Factors, Divisibility, Squares and Cubes

📚 Worked Examples

Study each example carefully, then try the practice questions!

Example 1: Prime Factorisation using a Factor Tree

Find the prime factorisation of 60.

Start: 60
Step 1: 60 = 2 × 30
Step 2: 30 = 2 × 15
Step 3: 15 = 3 × 5  ← both prime ✓
Step 4: Collect all prime leaves: 2, 2, 3, 5
✅ 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

💡 You can start with different factor pairs (e.g. 60 = 6 × 10) and always get the same answer!

Example 2: Finding HCF and LCM

Find the HCF and LCM of 24 and 36.

Prime factorisation of 24: 2³ × 3
Prime factorisation of 36: 2² × 3²
HCF: take the LOWEST power of each common prime:
Both share 2 and 3 → 2min(3,2) × 3min(1,2) = 2² × 3 = 12
LCM: take the HIGHEST power of every prime:
2max(3,2) × 3max(1,2) = 2³ × 3² = 8 × 9 = 72
✅ HCF(24, 36) = 12  |  LCM(24, 36) = 72
Check: 24 ÷ 12 = 2 ✓   36 ÷ 12 = 3 ✓   72 ÷ 24 = 3 ✓   72 ÷ 36 = 2 ✓

Example 3: Divisibility Rules

Is 4,536 divisible by 2, 3, 4, and 9?

÷ 2: 4,536 is even (ends in 6) → YES ✓
÷ 3: digit sum = 4 + 5 + 3 + 6 = 18, and 18 ÷ 3 = 6 → YES ✓
÷ 4: last two digits = 36, and 36 ÷ 4 = 9 → YES ✓
÷ 9: digit sum = 18, and 18 ÷ 9 = 2 → YES ✓
✅ 4,536 is divisible by 2, 3, 4 and 9.

Example 4: Squares, Cubes and Roots

Calculate: (a) 12²   (b) 4³   (c) √169   (d) ∛216

(a) 12² = 12 × 12 = 144
(b) 4³ = 4 × 4 × 4 = 16 × 4 = 64
(c) √169: What number × itself = 169? → 13 × 13 = 169 → √169 = 13
(d) ∛216: What number × itself × itself = 216? → 6 × 6 × 6 = 216 → ∛216 = 6
✅ Answers: (a) 144   (b) 64   (c) 13   (d) 6

Collins Stage 6 Cambridge Primary Maths — Unit 4

🔬 Factor Tree Builder

Enter a number and click Build Tree to see its prime factorisation step by step — plus all its factors!

Collins Stage 6 Cambridge Primary Maths — Unit 4

🎯 Drag Exercise 1: Is It Prime?

Which of these numbers is PRIME?

Drag the correct number into the drop zone below.

💡 Hint: A prime number has exactly 2 factors: 1 and itself. Check each number!

15 = 3 × 5  |  21 = 3 × 7  |  29 = ?  |  35 = 5 × 7

15
21
29
35

Drop the prime number here:

Prime number →
?

Collins Stage 6 Cambridge Primary Maths — Unit 4

🎯 Drag Exercise 2: Highest Common Factor

What is the Highest Common Factor (HCF) of 18 and 24?

Drag the correct answer into the drop zone below.

💡 Hint:

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Which is the highest factor that appears in both lists?

3
6
9
12

HCF(18, 24) =

?

Collins Stage 6 Cambridge Primary Maths — Unit 4

🎯 Drag Exercise 3: Lowest Common Multiple

What is the Lowest Common Multiple (LCM) of 4 and 6?

Drag the correct answer into the drop zone below.

💡 Hint:

Multiples of 4: 4, 8, 12, 16, 20, 24…

Multiples of 6: 6, 12, 18, 24…

What is the first (smallest) multiple that appears in both lists?

12
18
24
36

LCM(4, 6) =

?

Collins Stage 6 Cambridge Primary Maths — Unit 4

🎯 Drag Exercise 4: Square Root

What is √144?

Drag the correct answer into the drop zone below.

💡 Hint: Which number multiplied by itself gives 144?

11 × 11 = 121  |  12 × 12 = ?  |  13 × 13 = 169  |  14 × 14 = 196

11
12
13
14

√144 =

?

Collins Stage 6 Cambridge Primary Maths — Unit 4

🎯 Drag Exercise 5: Prime Factorisation

Which shows the correct prime factorisation of 36?

Drag the correct expression into the drop zone below.

💡 Hint: 36 = 4 × 9 = 2 × 2 × 3 × 3. Write using index (power) notation.

Remember: index notation means writing repeated multiplication as a power, e.g. 2 × 2 = 2²

2²×3²
2×3²
2³×3
4×9

36 =

?

Collins Stage 6 Cambridge Primary Maths — Unit 4

🎯 Drag Exercise 6: Cube Numbers

Which of these is a cube number?

Drag the correct answer into the drop zone below.

💡 Hint: Cube numbers are made by multiplying a number by itself three times.

1³ = 1  |  2³ = 8  |  3³ = 27  |  4³ = 64  |  5³ = 125

16
27
36
48

The cube number is:

?

Collins Stage 6 Cambridge Primary Maths — Unit 4

✏️ Practice Questions

Answer all 20 questions, then reveal the answers to check your work.

  1. List all the factors of 30.
  2. List the first 6 multiples of 7.
  3. Is 91 a prime number? Explain how you know.
  4. Find the prime factorisation of 48. Write your answer in index form.
  5. Find the Highest Common Factor (HCF) of 20 and 30.
  6. Find the Lowest Common Multiple (LCM) of 5 and 8.
  7. Calculate 7².
  8. Calculate 3³.
  9. Find √81.
  10. Find ∛27.
  11. Is 126 divisible by 3? Show how you know using the digit sum rule.
  12. Is 364 divisible by 4? Show how you know using the last-two-digits rule.
  13. Find the HCF of 36 and 48.
  14. Find the LCM of 6 and 9.
  15. List all prime numbers between 20 and 40.
  16. What is the prime factorisation of 100? Write in index form.
  17. Find the HCF of 75 and 100.
  18. What is ∛1000?
  19. Which is larger: 5³ or 10²? Show your working.
  20. Find the LCM of 8 and 12.

✅ Answers

  1. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
  2. First 6 multiples of 7: 7, 14, 21, 28, 35, 42
  3. No, 91 is not prime. 91 = 7 × 13 (it has more than 2 factors).
  4. 48 = 2 × 24 = 2 × 2 × 12 = 2 × 2 × 2 × 6 = 2 × 2 × 2 × 2 × 3 → 2⁴ × 3
  5. Factors of 20: 1,2,4,5,10,20. Factors of 30: 1,2,3,5,6,10,15,30. Highest in common: 10
  6. Multiples of 5: 5,10,15,20,25,30,35,40. Multiples of 8: 8,16,24,32,40. First in common: 40
  7. 7² = 7 × 7 = 49
  8. 3³ = 3 × 3 × 3 = 27
  9. √81 = 9 (since 9 × 9 = 81)
  10. ∛27 = 3 (since 3 × 3 × 3 = 27)
  11. Digit sum of 126: 1 + 2 + 6 = 9. 9 ÷ 3 = 3. Yes, 126 is divisible by 3.
  12. Last two digits of 364: 64. 64 ÷ 4 = 16. Yes, 364 is divisible by 4.
  13. 36 = 2² × 3²; 48 = 2⁴ × 3. HCF = 2² × 3 = 12
  14. Multiples of 6: 6,12,18. Multiples of 9: 9,18. First in common: 18
  15. Prime numbers between 20 and 40: 23, 29, 31, 37
  16. 100 = 10 × 10 = (2 × 5) × (2 × 5) = 2² × 5²
  17. 75 = 3 × 5²; 100 = 2² × 5². HCF = 5² = 25
  18. ∛1000 = 10 (since 10 × 10 × 10 = 1000)
  19. 5³ = 125; 10² = 100. 5³ is larger (125 > 100).
  20. Multiples of 8: 8,16,24. Multiples of 12: 12,24. First in common: 24

Collins Stage 6 Cambridge Primary Maths — Unit 4

🏆 Challenge Problems

These problems are harder — they combine ideas and require careful reasoning. Try each one before revealing the answers!

  1. Tom is planting trees in rows. He has 36 oak trees and 48 pine trees. Each row must contain only one type of tree, and every row must have the same number of trees. What is the maximum number of trees per row?
  2. Bus A leaves the station every 8 minutes. Bus B leaves the station every 12 minutes. They both leave together at 9:00 am. When is the next time they leave the station at the same time?
  3. A number has prime factorisation 2³ × 3 × 5. What is the number? How many factors does it have? (Hint: use the factor count formula.)
  4. Is 1,764 a perfect square? If so, find its square root. Show your working using prime factorisation.
  5. Find two prime numbers that add up to 50. How many different pairs can you find?
  6. The LCM of two numbers is 60 and their HCF is 4. If one of the numbers is 12, find the other number.
  7. A square garden has an area of 225 m². What is the length of one side of the garden?
  8. What is the smallest number that is divisible by both 12 and 18, and is also a perfect square?
  9. Write 360 as a product of prime factors in index form. Use the result to find the total number of factors that 360 has.
  10. I am thinking of a number. I am a cube number less than 200. I am also even. My cube root is a prime number. What number am I?

✅ Challenge Answers

  1. We need the largest number that divides into both 36 and 48 exactly.
    HCF(36, 48): 36 = 2² × 3²; 48 = 2⁴ × 3. HCF = 2² × 3 = 12.
    Maximum trees per row = 12.
  2. We need the smallest time that is a multiple of both 8 and 12.
    LCM(8, 12): 8 = 2³; 12 = 2² × 3. LCM = 2³ × 3 = 24 minutes.
    Next time together = 9:00 am + 24 minutes = 9:24 am.
  3. Number = 2³ × 3 × 5 = 8 × 3 × 5 = 120.
    Factor count formula: (3+1)(1+1)(1+1) = 4 × 2 × 2 = 16 factors.
  4. 1,764 = 4 × 441 = 4 × 9 × 49 = 2² × 3² × 7².
    Every prime appears to an even power → Yes, 1,764 is a perfect square.
    √1,764 = 2 × 3 × 7 = 42.
  5. Pairs of primes that sum to 50 (note: one must be 2 for the sum to be even… but 2 + 48 = 50, and 48 is not prime. So both primes must be odd, meaning both end in an odd digit):
    3 + 47 = 50 ✓  |  7 + 43 = 50 ✓  |  13 + 37 = 50 ✓  |  19 + 31 = 50 ✓
    4 pairs in total.
  6. Key relationship: Product of two numbers = LCM × HCF.
    Product = 60 × 4 = 240.
    Other number = 240 ÷ 12 = 20.
  7. Area = side² → side = √225.
    15 × 15 = 225 → Side length = 15 m.
  8. LCM(12, 18): 12 = 2² × 3; 18 = 2 × 3². LCM = 2² × 3² = 36.
    Is 36 a perfect square? 36 = 6² → Yes! The answer is 36.
  9. 360 = 36 × 10 = (4 × 9) × (2 × 5) = 2² × 3² × 2 × 5 = 2³ × 3² × 5.
    Number of factors = (3+1)(2+1)(1+1) = 4 × 3 × 2 = 24 factors.
  10. We need: a cube number, even, less than 200, with a prime cube root.
    Cube numbers less than 200: 1, 8, 27, 64, 125.
    Even ones: 8, 64.
    Cube root of 8 = 2 (prime ✓); Cube root of 64 = 4 (not prime ✗).
    The number is 8.

Collins Stage 6 Cambridge Primary Maths — Unit 4