1. What is a factor?

A factor of a number is a whole number that divides that number exactly (no remainder). For example: 3 is a factor of 12 because 12 ÷ 3 = 4 exactly.

2. Factor pairs

Factors often come in pairs: a × b = n. For n = 12, factor pairs are:

So the factors of 12 are: 1, 2, 3, 4, 6, 12 (written in order).

3. Prime and composite numbers

• Prime number: has exactly two different factors: 1 and itself. Example: 5 (factors 1,5).
• Composite number: has more than two factors. Example: 8 (factors 1,2,4,8).
• 1 is neither prime nor composite.

4. How to find factors (simple methods)

1. Start with 1 and the number itself — they are always factors.
2. Try dividing the number by integers 2, 3, 4, ... up to √n. If division has no remainder, both divisor and quotient are factors.
3. List each factor pair once (stop when divisor > quotient).

Example: Find factors of 30

Check 1 → 30 (yes), 2 → 15 (yes), 3 → 10 (yes), 4 → not exact, 5 → 6 (yes). Stop when divisor > quotient (5 ≥ 6? no, next 6 we already have). Factors: 1,2,3,5,6,10,15,30.

5. Prime factorisation (useful to find factors)

Break a number into prime numbers multiplied together. You can use a factor tree.

84
├─ 2 × 42
│   └─ 2 × 21
│       └─ 3 × 7
So 84 = 2 × 2 × 3 × 7 = 2^2 × 3 × 7
      

Once you have prime factors, you can list all factors by multiplying combinations of primes. Example: from 84 = 2^2 × 3 × 7, some factors are 1, 2, 4, 3, 6, 7, 12, 14, 21, 28, 42, 84.

6. Common factors and Highest Common Factor (HCF)

Common factors of two numbers are numbers that divide both exactly. The greatest of these is the HCF (also called GCF).

Method 1 – List factors: Find factors of each and pick the largest common one.

Method 2 – Using prime factorisation: Multiply the shared prime powers.

48  = 2^4 × 3
180 = 2^2 × 3^2 × 5
Common primes: 2^(min(4,2)) = 2^2, 3^(min(1,2)) = 3^1
HCF = 2^2 × 3 = 4 × 3 = 12
      

7. Quick divisibility tests (helps find factors faster)

• Divisible by 2: last digit even (0,2,4,6,8)
• Divisible by 3: sum of digits divisible by 3
• Divisible by 5: last digit 0 or 5
• Divisible by 9: sum of digits divisible by 9
• Divisible by 10: last digit 0

8. Practice questions

1. List all factors of 36.
2. Is 29 prime or composite? Explain.
3. Find the prime factorisation of 90.
4. Find the HCF of 54 and 72.
5. Give two numbers between 20 and 30 that are factors of 120.

9. Answers

1. Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.
2. 29 is prime — only factors are 1 and 29 (no other integer divides 29 exactly).
3. 90 = 2 × 45 = 2 × 3 × 15 = 2 × 3 × 3 × 5 = 2 × 3^2 × 5.
4. 54 = 2 × 3^3; 72 = 2^3 × 3^2. Common: 2^(min(1,3))=2^1, 3^(min(3,2))=3^2 -> HCF = 2 × 9 = 18.
5. Factors of 120 include 24 and 20 (both between 20 and 30 and both divide 120). Other choices: 21 is not, 25 is not.