HCF & LCM Calculator: Step-by-Step Prime Factorisation Working
Interactive calculator to find HCF, LCM, and all factors of numbers with step-by-step prime factorisation method for Secondary 1-2 Math.
HCF & LCM Calculator
Find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of any numbers with clear, step-by-step prime factorisation working — perfect for checking your Secondary 1-2 Math homework!
Try the Calculator
Enter 2 or 3 numbers to find their HCF, LCM, or all factors. The calculator shows the complete prime factorisation method that Singapore Math teachers expect to see in your working.
HCF & LCM Calculator
Find Highest Common Factor, Lowest Common Multiple, or all factors with step-by-step working
Understanding HCF and LCM
HCF and LCM are fundamental concepts in the Factors & Multiples topic, typically covered in Secondary 1 Math in Singapore. Let’s break down what each one means and when to use them.
What is HCF (Highest Common Factor)?
The Highest Common Factor (HCF), also called Greatest Common Divisor (GCD), is the largest number that divides two or more numbers exactly (with no remainder).
💡 When to Use HCF
Use HCF when you need to divide things into equal groups or simplify fractions:
- Sharing items equally among people
- Reducing fractions to lowest terms
- Finding the largest square tile that fits a room perfectly
Example: Find the HCF of 24 and 36
| Step | Working |
|---|---|
| 1. Prime factorise 24 | 24 = 2 × 2 × 2 × 3 |
| 2. Prime factorise 36 | 36 = 2 × 2 × 3 × 3 |
| 3. Find common factors | Both have: 2, 2, and 3 |
| 4. Multiply common factors | HCF = 2 × 2 × 3 = 12 |
What is LCM (Lowest Common Multiple)?
The Lowest Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers.
💡 When to Use LCM
Use LCM when you need to find when events coincide or add fractions with different denominators:
- Two buses leave at different intervals — when do they leave together again?
- Finding a common denominator for fractions
- Calculating when rotating patterns align
Example: Find the LCM of 24 and 36
| Step | Working |
|---|---|
| 1. Prime factorise 24 | 24 = 2³ × 3 |
| 2. Prime factorise 36 | 36 = 2² × 3² |
| 3. Take highest power of each prime | 2³ and 3² |
| 4. Multiply highest powers | LCM = 2³ × 3² = 8 × 9 = 72 |
The Prime Factorisation Method
The prime factorisation method is the standard approach taught in Singapore secondary schools. Here’s how it works:
Step 1: Prime Factorise Each Number
Use the repeated division method:
- Divide by the smallest prime (2) as many times as possible
- Move to the next prime (3, 5, 7…) when division is no longer exact
- Continue until you reach 1
Prime Factorisation of 120
120 ÷ 2 = 60
60 ÷ 2 = 30
30 ÷ 2 = 15
15 ÷ 3 = 5
5 ÷ 5 = 1 ✓
Therefore: 120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5
Step 2: Compare Prime Factors
For HCF: Take common factors at their lowest power For LCM: Take all factors at their highest power
⚠️ Common Mistake Alert
Students often confuse which powers to use:
- HCF = Common primes × LOWEST power (think: what’s definitely shared)
- LCM = All primes × HIGHEST power (think: what covers everything)
The Quick Check Formula
For two numbers only, there’s a useful relationship:
Example: For 24 and 36:
- HCF = 12
- LCM = 72
- Check: 12 × 72 = 864 = 24 × 36 ✓
💡 Use This to Find LCM Quickly
If you know the HCF, you can find the LCM:
For 24 and 36: LCM = (24 × 36) ÷ 12 = 864 ÷ 12 = 72
Worked Examples
Example 1: Finding HCF and LCM of Three Numbers
Find the HCF and LCM of 12, 18, and 30
Step 1: Prime Factorise
12 = 2² × 3
18 = 2 × 3²
30 = 2 × 3 × 5
Step 2: Find HCF
Common primes: 2 and 3
Lowest powers: 2¹ and 3¹
HCF = 2 × 3 = 6
Step 3: Find LCM
All primes: 2, 3, and 5
Highest powers: 2², 3², and 5¹
LCM = 4 × 9 × 5 = 180
Example 2: Real-World Application (HCF)
Tiling Problem
Problem:
A rectangular floor measures 120 cm by 150 cm. What is the largest square tile that can be used to cover the floor completely without cutting?
Solution:
We need the largest number that divides BOTH 120 and 150 exactly — that’s the HCF!
120 = 2³ × 3 × 5
150 = 2 × 3 × 5²
HCF = 2¹ × 3¹ × 5¹ = 30
Answer: The largest square tile is 30 cm × 30 cm
Example 3: Real-World Application (LCM)
Bus Schedule Problem
Problem:
Bus A leaves the terminal every 12 minutes. Bus B leaves every 18 minutes. If both buses leave together at 8:00 AM, when will they next leave together?
Solution:
We need the smallest time that is a multiple of BOTH 12 and 18 — that’s the LCM!
12 = 2² × 3
18 = 2 × 3²
LCM = 2² × 3² = 4 × 9 = 36
Answer: They will next leave together at 8:36 AM (36 minutes later)
Common Mistakes to Avoid
| Mistake | Why It’s Wrong | Correct Approach |
|---|---|---|
| Using highest powers for HCF | This gives LCM, not HCF | Use LOWEST powers of COMMON factors |
| Using lowest powers for LCM | This gives HCF, not LCM | Use HIGHEST powers of ALL factors |
| Forgetting a prime factor | Missing factors in one number | List ALL primes from BOTH numbers for LCM |
| Including non-common factors in HCF | HCF only uses shared factors | Only include primes that appear in ALL numbers |
Quick Reference: HCF vs LCM
| HCF | LCM | |
|---|---|---|
| Full name | Highest Common Factor | Lowest Common Multiple |
| What it finds | Largest shared divisor | Smallest shared multiple |
| Prime factor rule | Common primes, LOWEST power | All primes, HIGHEST power |
| Use when… | Dividing into equal groups | Finding when events coincide |
| Example use | Simplifying fractions | Adding fractions |
| Result is… | ≤ smallest number | ≥ largest number |
Practice Using the Calculator
Try these practice questions with the calculator above:
- Easy: Find HCF and LCM of 8 and 12
- Medium: Find HCF and LCM of 24, 36, and 48
- Challenge: Find all factors of 120
💡 Study Tip
After using the calculator, try solving similar problems WITHOUT it. Use the calculator to check your working — this builds exam confidence!
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