Number Theory

Factors and Multiples

Factors are whole numbers that divide exactly into a given number with no remainder.

To find all factors of 12, check which numbers divide into it evenly:

• 1 × 12 = 12
• 2 × 6 = 12
• 3 × 4 = 12
• Factors of 12: 1, 2, 3, 4, 6, 12

Multiples are the results of multiplying a number by positive integers (1, 2, 3, 4…):

• Multiples of 4: 4, 8, 12, 16, 20, 24, …
• Multiples of 7: 7, 14, 21, 28, 35, 42, …

Key Terms

• Factor — a number that divides exactly into another (no remainder)
• Multiple — the product of a number and a positive integer
• Prime — a number with exactly two factors: 1 and itself
• Composite — a number with more than two factors

Prime and Composite Numbers

A prime number has exactly two factors: 1 and itself. It cannot be divided evenly by any other whole number.

First 10 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Important facts:

• 1 is NOT prime — it has only one factor (itself), not two.
• 2 is the only even prime — all other even numbers have 2 as a factor, so they have more than two factors.

A composite number has more than two factors:

• 4 (factors: 1, 2, 4) — composite
• 9 (factors: 1, 3, 9) — composite
• 12 (factors: 1, 2, 3, 4, 6, 12) — composite

Zero and 1 are neither prime nor composite.

Prime Factorisation

Every composite number can be expressed as a unique product of prime numbers. This is called the prime factorisation or prime factor form.

Use a factor tree to break the number down step by step:

Worked Example: Find the prime factorisation of 36.

• 36 = 2 × 18
• 18 = 2 × 9
• 9 = 3 × 3
• So: 36 = 2 × 2 × 3 × 3 = 2² × 3²

Worked Example: Find the prime factorisation of 60.

• 60 = 2 × 30
• 30 = 2 × 15
• 15 = 3 × 5
• So: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

No matter which factors you start with, you always reach the same prime factorisation (this is the Fundamental Theorem of Arithmetic).

Greatest Common Factor (GCF)

The GCF (also called Highest Common Factor or HCF) is the largest factor shared by two or more numbers. It is useful for simplifying fractions and dividing quantities evenly.

Method 1: List factors

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

Method 2: Prime factorisation

• 18 = 2 × 3²
• 24 = 2³ × 3
• Common primes, lowest powers: 2¹ × 3¹ = 6

Worked Example: Simplify 18/24 using the GCF. GCF = 6. So 18/24 = (18÷6)/(24÷6) = 3/4

Lowest Common Multiple (LCM)

The LCM is the smallest multiple shared by two or more numbers. It is essential for adding and subtracting fractions with different denominators.

Method 1: List multiples

LCM of 4 and 6:

• Multiples of 4: 4, 8, 12, 16, 20, 24
• Multiples of 6: 6, 12, 18, 24
• LCM = 12

Method 2: Prime factorisation

• 4 = 2²
• 6 = 2 × 3
• Take each prime factor at its highest power: 2² × 3 = 12

Worked Example: Add 1/4 + 1/6 using the LCM. LCM of 4 and 6 = 12. So: 3/12 + 2/12 = 5/12

Divisibility Rules

Quick tests that tell you whether a number is divisible by another without dividing:

Divisible by	Rule	Example
2	Last digit is even	348 ✓ (8 is even)
3	Sum of digits divisible by 3	27: 2+7=9 ✓
4	Last two digits divisible by 4	312: 12÷4=3 ✓
5	Last digit is 0 or 5	435 ✓
6	Divisible by both 2 and 3	18: even + 1+8=9 ✓
9	Sum of digits divisible by 9	81: 8+1=9 ✓
10	Last digit is 0	270 ✓

Worked Example: Is 486 divisible by 6?

• By 2: last digit is 6 (even) ✓
• By 3: 4+8+6 = 18, and 18 ÷ 3 = 6 ✓
• Divisible by both 2 and 3 → yes, divisible by 6

Common Mistakes

• Mistake: including 1 as a prime number. Fix: 1 has only one factor (itself), not two. Prime numbers need exactly two distinct factors.
• Mistake: stopping the factor tree too early (e.g. leaving 6 in the tree instead of breaking it into 2 × 3). Fix: keep splitting until all branches end in primes.
• Mistake: confusing GCF and LCM. Fix: GCF is the largest shared factor (smaller or equal to both numbers); LCM is the smallest shared multiple (larger or equal to both).
• Mistake: assuming the LCM is always the product of the two numbers. Fix: LCM of 4 and 6 is 12, not 24. Use prime factorisation to find the true LCM.

Tips and Tricks

• A quick check for primes: test divisibility by all primes up to the square root of the number. If none divide it, it is prime.
• To find the GCF and LCM together using prime factorisation: GCF uses the lowest power of each shared prime; LCM uses the highest power of each prime.
• Divisibility by 6 = divisible by 2 AND by 3 — you need both.
• Factor pairs come in two: once you find one factor, divide to get its partner (e.g. 3 is a factor of 12, so 12÷3=4 is also a factor).