What this chapter is about
This chapter is about special whole numbers called prime numbers and their cousins, composite numbers. You will learn how to tell if a number is prime or composite by looking at its factors.
You will also learn how to break any number into a product of primes. This is called prime factorisation. It is like finding the building blocks of a number. These ideas help you work with factors, multiples, divisibility and many real-life problems later.
By the end, you should be able to list factors of a number, decide if it is prime or composite, and write it as a product of prime numbers.
Key ideas
- A factor of a number divides it exactly with no remainder. For example, factors of 12 are 1, 2, 3, 4, 6 and 12.
- A prime number has exactly two factors: 1 and itself. Examples: 2, 3, 5, 7, 11, 13.
- A composite number has more than two factors. Examples: 4, 6, 8, 9, 10, 12.
- The number 1 is special. It has only one factor (itself), so it is neither prime nor composite.
- 2 is the smallest prime and the only even prime number. Every other even number is composite.
- Prime factorisation means writing a number as a product of primes. For example, 18 = 2 × 3 × 3.
- Every composite number can be written as a product of primes in only one way (ignoring order). This is called the Fundamental Theorem of Arithmetic.
- To test if a small number is prime, check whether any prime less than it divides it exactly.
Formulas and facts to remember
- Finding factors: Divide the number by 1, 2, 3, … and list all divisors that give remainder 0.
- Prime check rule: If no number from 2 up to just below the number divides it, the number is prime.
- Prime factorisation by division: Keep dividing by the smallest prime that works until you reach 1. Collect all the primes used.
- Useful small primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
- Divisibility shortcut for 2: A number ending in 0, 2, 4, 6 or 8 is divisible by 2.
- Divisibility shortcut for 3: If the sum of digits is divisible by 3, the number is divisible by 3.
Worked examples
Example 1: Is 29 a prime number?
Step 1: Try dividing 29 by 2. 29 ÷ 2 = 14 remainder 1. Not divisible. Step 2: Try 3. 29 ÷ 3 = 9 remainder 2. Not divisible. Step 3: Try 5. 29 ÷ 5 = 5 remainder 4. Not divisible. Step 4: Try 7. 29 ÷ 7 = 4 remainder 1. Not divisible. No prime up to 7 divides 29. So 29 is a prime number.
Example 2: Write the prime factorisation of 60.
Step 1: 60 ÷ 2 = 30. Write down 2. Step 2: 30 ÷ 2 = 15. Write down another 2. Step 3: 15 ÷ 3 = 5. Write down 3. Step 4: 5 ÷ 5 = 1. Write down 5. Collect all primes: 60 = 2 × 2 × 3 × 5.
Example 3: List all factors of 28.
Divide 28 by 1, 2, 3, … up to 28. 28 ÷ 1 = 28 ✓ 28 ÷ 2 = 14 ✓ 28 ÷ 4 = 7 ✓ 28 ÷ 7 = 4 ✓ 28 ÷ 14 = 2 ✓ 28 ÷ 28 = 1 ✓ Factors of 28: 1, 2, 4, 7, 14, 28. Since 28 has more than two factors, it is composite.
Common mistakes
- Thinking 1 is a prime number → Remember, 1 has only one factor, so it is neither prime nor composite.
- Forgetting that 2 is prime because it is even → 2 is the only even prime; all other even numbers are composite.
- Stopping prime factorisation too early (e.g., writing 12 = 3 × 4) → Keep going: 4 = 2 × 2, so 12 = 2 × 2 × 3.
- Confusing factors and multiples → Factors divide the number; multiples are the number times 1, 2, 3, …
- Missing factor pairs (e.g., listing only small factors) → Check pairs: if 2 divides 28, then 28 ÷ 2 = 14 is also a factor.
Quick revision
- A prime has exactly two factors: 1 and itself.
- A composite has more than two factors.
- 1 is neither prime nor composite.
- 2 is the smallest and only even prime.
- Prime factorisation breaks a number into a product of primes.
- Always divide by the smallest prime first when factorising.