Chapter focus: Numbers can be explored through their properties and relationships. In this chapter, students develop number sense by investigating factors, multiples, divisibility and patterns, and by explaining why a statement about numbers is true or false.

🎯 Learning Goals

• Understand factors and multiples of a number.
• Recognise common factors and common multiples.
• Use divisibility rules to test numbers efficiently.
• Identify prime and composite numbers.
• Find prime factors of numbers.
• Use number properties to solve puzzles and reasoning problems.

🔢 1. Factors

A factor of a number is a whole number that divides the number exactly, leaving no remainder.

For example, the factors of 12 are 1, 2, 3, 4, 6 and 12 because each of these numbers divides 12 exactly.

How to Find Factors

1. Start with 1. Every positive whole number has 1 as a factor.
2. Check numbers that divide the given number exactly.
3. Record the matching factor pairs.
4. Stop once the pairs begin repeating in reverse order.

✖️ 2. Multiples

A multiple of a number is obtained by multiplying it by a whole number.

Multiples of 5 are 5, 10, 15, 20, 25, 30, …

Every number is a multiple of 1, and every number is a multiple of itself.

🔁 Factors vs Multiples

🧩 3. Divisibility

Divisibility rules help us decide quickly whether one number divides another number exactly.

⭐ 4. Prime and Composite Numbers

A prime number has exactly two positive factors: 1 and itself.

Examples: 2, 3, 5, 7, 11, 13, 17.

A composite number has more than two positive factors. Examples include 4, 6, 8, 9, 10 and 12.

🌳 5. Prime Factorisation

Prime factorisation means expressing a number as a product of prime numbers.

For example:

60 = 2 × 2 × 3 × 5

All the factors on the right are prime.

🔗 6. Common Factors and Common Multiples

Factors shared by two or more numbers are called common factors.

For 12 and 18:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• Common factors: 1, 2, 3, 6

Numbers that occur in the multiples of two or more numbers are common multiples.

Multiples of 4: 4, 8, 12, 16, 20, 24, …

Multiples of 6: 6, 12, 18, 24, …

Common multiples include 12, 24, …

📋 Worked Examples

Example 1 – Factors

Question: Find all factors of 24.

Factor pairs are 1 × 24, 2 × 12, 3 × 8 and 4 × 6.

Answer: 1, 2, 3, 4, 6, 8, 12 and 24.

Example 2 – Divisibility by 3

Question: Is 4,326 divisible by 3?

Add the digits: 4 + 3 + 2 + 6 = 15. Since 15 is divisible by 3, 4,326 is divisible by 3.

Example 3 – Prime or Composite?

Question: Is 29 prime?

The only positive factors of 29 are 1 and 29.

Answer: 29 is prime.

Example 4 – Prime Factorisation

Question: Write 36 as a product of prime numbers.

36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3.

🧠 Number Reasoning

1. Why is every factor of a number less than or equal to that number?
2. Why can a number have many factors but infinitely many multiples?
3. Why is 2 the only even prime number?
4. Can an odd number be divisible by 2? Explain.
5. If a number is divisible by 10, what can you say about its divisibility by 2 and 5?

📝 Important Practice Questions

1. Write all factors of 36.
2. Write the first ten multiples of 7.
3. Find the common factors of 16 and 24.
4. Find three common multiples of 4 and 6.
5. Use a divisibility rule to determine whether 3,426 is divisible by 2 and 3.
6. Classify 17, 21, 29 and 35 as prime or composite.
7. Explain why 1 is neither prime nor composite.
8. Write 48 as a product of prime numbers.
9. Give two numbers that have exactly three positive factors.
10. Give an example of two numbers with more than one common factor.

🔥 Competency / HOTS Questions

1. A number is divisible by both 2 and 5. What must be its last digit? Explain.
2. Riya says that every multiple of a number is greater than the number. Is this always true? Consider the first multiple carefully.
3. Can two prime numbers have a common factor other than 1? Explain.
4. A number has exactly two positive factors. What type of number is it? Why?
5. Find the smallest positive number that is a multiple of both 6 and 8. Show your reasoning.

🧩 MCQ Practice

1. Which is a factor of 24?
   A. 5   B. 7   C. 6   D. 10
2. Which is a multiple of 9?
   A. 28   B. 36   C. 41   D. 50
3. Which number is prime?
   A. 21   B. 27   C. 31   D. 39
4. Which number is divisible by 5?
   A. 42   B. 63   C. 75   D. 88
5. Which statement is correct?
   A. 1 is prime   B. 1 is composite   C. 1 is neither prime nor composite   D. 1 is both
6. Which is a factor pair of 36?
   A. 5 and 7   B. 4 and 9   C. 3 and 11   D. 2 and 20
7. A number ending in 0 is always divisible by —
   A. 2 and 5   B. 3 only   C. 9 only   D. 7 only
8. Which is the prime factorisation of 18?
   A. 2 × 9   B. 3 × 6   C. 2 × 3 × 3   D. 1 × 18

Answer Key: 1-C, 2-B, 3-C, 4-C, 5-C, 6-B, 7-A, 8-C

📄 Printable Worksheet

1. List all factors of 30.
2. Write the first eight multiples of 6.
3. Circle the prime numbers: 11, 15, 17, 21, 23, 25, 29.
4. Use divisibility rules to test 4,560 for divisibility by 2, 5 and 10.
5. Find the common factors of 18 and 27.
6. Find four common multiples of 3 and 5.
7. Write 72 as a product of prime numbers.
8. Explain why 2 is prime but 4 is composite.
9. Find the smallest number greater than 20 that is divisible by both 4 and 6.
10. Create your own number puzzle involving factors or multiples and solve it.

🔁 Quick Revision Table

✅ Chapter Mastery Checklist

• ☐ I can find factors and factor pairs.
• ☐ I can list multiples of a number.
• ☐ I can distinguish factors from multiples.
• ☐ I can use common divisibility rules.
• ☐ I can identify prime and composite numbers.
• ☐ I can write prime factorisations.
• ☐ I can find common factors and common multiples.
• ☐ I can explain my reasoning in number puzzles.

🔗 Continue Learning

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