Chapter focus: Prime numbers are the building blocks of whole numbers. This chapter develops students’ ability to recognise primes, use divisibility, find factors and prime factorisations, and reason about number relationships.

🎯 Learning Goals

• Understand prime and composite numbers.
• Recognise factors and factor pairs.
• Use divisibility tests efficiently.
• Find prime factors and prime factorisation.
• Identify common factors and common multiples.
• Use number properties to solve puzzles and explain reasoning.

⭐ 1. Prime Numbers

A prime number has exactly two positive factors: 1 and itself. Examples include 2, 3, 5, 7, 11 and 13.

2 is the only even prime number. Every other even number has 2 as a factor as well as 1 and itself, so it has more than two factors.

🔢 2. Composite Numbers

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

✖️ 3. Factors and Factor Pairs

A factor divides a number exactly. Factor pairs multiply to give the number.

For 36: 1×36, 2×18, 3×12, 4×9 and 6×6.

🧩 4. Divisibility Rules

🌳 5. Prime Factorisation

Prime factorisation expresses a number as a product of prime numbers.

Example: 84 = 2 × 2 × 3 × 7.

You can find prime factors by repeatedly dividing by the smallest possible prime factor until only primes remain.

🔗 6. Common Factors and Common Multiples

Common factors are factors shared by two or more numbers. Common multiples appear in the multiplication lists of two or more numbers.

For 12 and 18, common factors are 1, 2, 3 and 6. Common multiples include 36, 72, 108, …

🧠 7. Prime Number Reasoning

1. Why is 2 the only even prime?
2. Why is 1 neither prime nor composite?
3. Can two different prime numbers have a common factor greater than 1?
4. How can divisibility rules save time?
5. Why does every composite number have at least one factor other than 1 and itself?

📋 Worked Examples

Example 1 – Prime or Composite?

Question: Is 37 prime?

Numbers between 1 and 37 must be checked for factors. 37 is not divisible by 2, 3 or 5, and no smaller prime divides it.

Answer: 37 is prime.

Example 2 – Divisibility by 9

Is 5,472 divisible by 9? Add the digits: 5+4+7+2 = 18. Since 18 is divisible by 9, 5,472 is divisible by 9.

Example 3 – Prime Factorisation

60 = 2×30 = 2×2×15 = 2×2×3×5.

Example 4 – Common Factor

Factors of 20 are 1, 2, 4, 5, 10, 20. Factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. Common factors are 1, 2, 5 and 10.

📝 Important Practice Questions

1. Define a prime number and give five examples.
2. Define a composite number and give five examples.
3. Explain why 1 is neither prime nor composite.
4. List all factors of 48.
5. Use divisibility rules to test 3,240 for 2, 3, 5, 9 and 10.
6. Write 72 as a product of prime numbers.
7. Find the common factors of 24 and 36.
8. Find three common multiples of 8 and 12.
9. Explain why every number greater than 1 is either prime or composite.
10. Write two numbers whose only common factor is 1.

🔥 Competency / HOTS Questions

1. A number is divisible by both 2 and 5. What must its last digit be? Explain.
2. A student says 91 is prime because it is odd. Identify the error and test 91.
3. Find the smallest composite number and explain why no smaller number greater than 1 is composite.
4. Can the sum of two prime numbers be odd? Give an example and explain.
5. A number has exactly three positive factors. What can you say about its form? Give an example.

🧩 MCQ Practice

1. Which is prime? A. 21 B. 29 C. 35 D. 39
2. Which number is composite? A. 13 B. 17 C. 21 D. 23
3. Which is the only even prime number? A. 0 B. 1 C. 2 D. 4
4. Which number is divisible by 9? A. 124 B. 234 C. 245 D. 316
5. Which is a factor pair of 48? A. 5, 9 B. 6, 8 C. 7, 8 D. 3, 20
6. 1 is — A. prime B. composite C. both D. neither
7. Prime factorisation of 24 is — A. 2×12 B. 3×8 C. 2×2×2×3 D. 1×24
8. A number ending in 0 is divisible by — A. 2 and 5 only B. 2, 5 and 10 C. 3 only D. 9 only

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

📄 Printable Worksheet

1. Circle the primes: 7, 12, 17, 19, 21, 31, 35.
2. Write all factors of 36.
3. Use divisibility rules for 2, 3, 5 and 9 on 4,590.
4. Find the prime factorisation of 48.
5. Find common factors of 18 and 30.
6. Write four common multiples of 4 and 6.
7. Explain why 2 is prime but 6 is composite.
8. Find the smallest prime greater than 30.
9. Find a composite number between 40 and 50 and explain why it is composite.
10. Create one prime-number puzzle for a classmate.

🔁 Quick Revision Table

✅ Chapter Mastery Checklist

• ☐ I can identify prime and composite numbers.
• ☐ I can explain why 1 is neither.
• ☐ I can find factors and factor pairs.
• ☐ I can use divisibility rules.
• ☐ I can find prime factorisations.
• ☐ I can identify common factors and multiples.
• ☐ I can explain my reasoning.

🔗 Continue Learning

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