Factors & Multiples

Factors

Factors of a number are the whole numbers that divide into it exactly, leaving no remainder.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

A quick way to find all factors is to list factor pairs — two numbers that multiply to give the target:

Factor pairs of 20: (1, 20), (2, 10), (4, 5) So the factors of 20 are: 1, 2, 4, 5, 10, 20

Every number has at least two factors: 1 and itself. Numbers with exactly two factors are prime numbers.

Multiples

Multiples are the results of multiplying a number by 1, 2, 3, 4, and so on — essentially the times table for that number.

Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56...

Unlike factors (which are finite and limited), a number has infinitely many multiples.

Prime Numbers

A prime number has exactly two factors: 1 and itself.

The first ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Important notes:

• 1 is not prime — it has only one factor.
• 2 is the only even prime — all other even numbers have 2 as a factor.

A composite number has more than two factors (e.g., 12 has six factors).

Prime Factorisation

Every composite number can be expressed as a unique product of prime numbers. This is found using a factor tree.

Example: Prime factorisation of 60

Split 60 into any factor pair, then continue splitting until all branches are prime:

• 60 = 4 × 15 = (2 × 2) × (3 × 5)
• 60 = 2² × 3 × 5

Example: Prime factorisation of 36

• 36 = 4 × 9 = (2 × 2) × (3 × 3)
• 36 = 2² × 3²

Highest Common Factor (HCF)

The HCF of two or more numbers is the largest factor they share.

Method 1 — List factors:

• Factors of 12: 1, 2, 3, 4, 6, 12
• Factors of 18: 1, 2, 3, 6, 9, 18
• HCF(12, 18) = 6

Method 2 — Prime factorisation:

• 12 = 2² × 3
• 18 = 2 × 3²
• Take the lowest power of each shared prime: 2¹ × 3¹ = 6

Lowest Common Multiple (LCM)

The LCM is the smallest multiple shared by two or more numbers.

Method 1 — List multiples:

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

Method 2 — Prime factorisation:

• 4 = 2²
• 6 = 2 × 3
• Take the highest power of each prime that appears: 2² × 3 = 12

Worked Example: HCF and LCM of 30 and 45

Prime factorisations:

• 30 = 2 × 3 × 5
• 45 = 3² × 5

HCF: take lowest powers of shared primes → 3 × 5 = 15 LCM: take highest powers of all primes → 2 × 3² × 5 = 90

When to Use HCF vs LCM

Problem type	Use
Splitting into equal groups	HCF
Simplifying fractions	HCF
Finding a common denominator	LCM
Adding/subtracting fractions	LCM
Scheduling / repeating events	LCM

Example (HCF): 24 red tiles and 36 blue tiles need to be arranged in equal rows with no tiles left over. The largest equal row size is HCF(24, 36) = 12.

Example (LCM): Bus A comes every 4 minutes, Bus B every 6 minutes. They both leave at 9:00. When do they next leave together? LCM(4, 6) = 12 → at 9:12.

Common Mistakes

• Confusing factors and multiples — factors are smaller than or equal to the number; multiples are larger than or equal to the number.
• Missing factor pairs — work systematically from 1 upward to avoid gaps.
• Including 1 as prime — 1 is not prime; it has only one factor.
• Using HCF when LCM is needed (or vice versa) — re-read the question carefully.

Tips and Tricks

• To find the HCF and LCM together efficiently, always use prime factorisation — one tree gives you both.
• A shortcut: HCF × LCM = product of the two numbers. Check: HCF(4,6) × LCM(4,6) = 2 × 12 = 24 = 4 × 6 ✓
• When simplifying a fraction, divide numerator and denominator by their HCF.
• If two numbers share no common prime factors, their HCF is 1 (they are called coprime) and their LCM is their product.