Number Theory
Factors and Multiples
Factors are numbers that divide exactly into a given number (no remainder).
Factors of 12: 1, 2, 3, 4, 6, 12
Multiples are the results of multiplying a number by 1, 2, 3, 4, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Prime and Composite Numbers
A prime number has exactly two factors: 1 and itself.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29 ...
Note: 1 is NOT prime (it has only one factor). 2 is the only even prime number.
A composite number has more than two factors.
4 (factors: 1, 2, 4) Β· 9 (factors: 1, 3, 9) Β· 12 (factors: 1, 2, 3, 4, 6, 12)
Prime Factorisation
Every composite number can be broken down into a product of prime numbers. Use a factor tree.
36 = 2 Γ 18 = 2 Γ 2 Γ 9 = 2 Γ 2 Γ 3 Γ 3 = 2Β² Γ 3Β²
Greatest Common Factor (GCF)
The GCF is the largest factor shared by two or more numbers. Useful for simplifying fractions.
GCF of 18 and 24: Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 GCF = 6
Lowest Common Multiple (LCM)
The LCM is the smallest multiple shared by two or more numbers. Useful for adding fractions.
LCM of 4 and 6: Multiples of 4: 4, 8, 12, 16, ... Multiples of 6: 6, 12, 18, ... LCM = 12
Divisibility Rules
Quick tests to check if a number divides evenly without doing the full calculation.
| Divisible by | Rule |
|---|---|
| 2 | Last digit is even (0, 2, 4, 6, 8) |
| 3 | Sum of digits is divisible by 3 |
| 4 | Last two digits form a number divisible by 4 |
| 5 | Last digit is 0 or 5 |
| 6 | Divisible by both 2 and 3 |
| 9 | Sum of digits is divisible by 9 |
| 10 | Last digit is 0 |
Is 27 divisible by 3? Sum of digits = 2 + 7 = 9. Yes, 9 Γ· 3 = 3. β