Factors and Multiples
Factor: A number that divides another number exactly (remainder = 0).
- Factors of 12: 1, 2, 3, 4, 6, 12
- Every number has 1 and itself as factors.
- A number has a finite (limited) number of factors.
Multiple: A number obtained by multiplying a given number by 1, 2, 3, …
- Multiples of 5: 5, 10, 15, 20, 25, …
- Every number has an infinite number of multiples.
- Every number is a multiple of itself.
Key link: If a is a factor of b, then b is a multiple of a. Example: 4 is a factor of 12 → 12 is a multiple of 4.
Prime and Composite Numbers
Prime number: Has exactly 2 factors — 1 and itself.
- Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Composite number: Has more than 2 factors.
- Examples: 4, 6, 8, 9, 10, 12, 14, 15, 16, …
Special cases:
- 1 is neither prime nor composite (only 1 factor).
- 2 is the only even prime number.
- All other even numbers are composite.
Sieve of Eratosthenes
Method to find all prime numbers up to a given number: 1. Write numbers 1 to 100. 2. Cross out 1. 3. Circle 2, cross out all multiples of 2. 4. Circle 3, cross out all multiples of 3. 5. Repeat for 5, 7, … 6. All circled numbers are prime.
Primes up to 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Divisibility Rules
| Divisor | Rule | Example |
|---|---|---|
| 2 | Last digit is 0, 2, 4, 6, or 8 | 348 → last digit 8 → divisible |
| 3 | Sum of digits divisible by 3 | 156 → 1+5+6=12 → 12÷3=4 → yes |
| 4 | Last two digits form a number divisible by 4 | 312 → 12÷4=3 → yes |
| 5 | Last digit is 0 or 5 | 235 → last digit 5 → yes |
| 6 | Divisible by both 2 and 3 | 318 → even + 3+1+8=12 → yes |
| 8 | Last three digits divisible by 8 | 1,024 → 024÷8=3 → yes |
| 9 | Sum of digits divisible by 9 | 729 → 7+2+9=18 → 18÷9=2 → yes |
| 10 | Last digit is 0 | 230 → yes |
| 11 | Difference of sum of digits in odd and even positions = 0 or 11 | 2,728 → (2+2)−(7+8) nope; need to check |
Prime Factorisation
Every composite number can be written as a product of prime factors.
Method — Factor Tree:
Method — Division:
HCF — Highest Common Factor
The largest factor common to two or more numbers.
Example: Find HCF of 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
- HCF = 6
Prime Factorisation Method:
- 12 = 2² × 3
- 18 = 2 × 3²
- HCF = 2¹ × 3¹ = 6 (take the lowest power of each common prime)
LCM — Lowest Common Multiple
The smallest multiple common to two or more numbers.
Example: Find LCM of 4 and 6.
- Multiples of 4: 4, 8, 12, 16, 20, 24, …
- Multiples of 6: 6, 12, 18, 24, …
- LCM = 12
Prime Factorisation Method:
- 4 = 2²
- 6 = 2 × 3
- LCM = 2² × 3 = 12 (take the highest power of each prime)
Relationship Between HCF and LCM
HCF × LCM = Product of the two numbers
Check: HCF(4,6) × LCM(4,6) = 2 × 12 = 24 = 4 × 6 ✅
Key Vocabulary
| Term | Meaning |
|---|---|
| Factor | Divides a number exactly |
| Multiple | Result of multiplying a number |
| Prime | Exactly 2 factors (1 and itself) |
| Composite | More than 2 factors |
| Co-prime | Two numbers with HCF = 1 |
| HCF | Highest Common Factor |
| LCM | Lowest Common Multiple |
| Prime factorisation | Expressing a number as product of primes |
Exam Tips
- 1 is neither prime nor composite — asked in 1-mark questions.
- 2 is the only even prime — very common mistake to say all primes are odd.
- Divisibility by 6 = divisible by 2 AND 3 (both conditions must hold).
- Divisibility by 9: if sum of digits is divisible by 9 → also divisible by 3 (but not vice versa).
- HCF × LCM = Product of two numbers — use this to find one when the other is given.
- Co-prime numbers: two numbers whose HCF = 1 (e.g., 8 and 15 are co-prime — HCF = 1).