LCM & HCF Calculator
Calculate the Lowest Common Multiple (LCM) and Highest Common Factor (HCF / GCF) for 2 to 6 numbers. Shows full prime factorisation working for each number, with the step-by-step method required in GCSE Maths.
What is LCM (Lowest Common Multiple)?
The Lowest Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all of those numbers. It is also called the Least Common Multiple. For example, LCM(4, 6) = 12 because 12 is the smallest number that appears in both the 4-times table (4, 8, 12, 16...) and the 6-times table (6, 12, 18, 24...).
What is HCF (Highest Common Factor)?
The Highest Common Factor (HCF) of two or more numbers is the largest positive integer that divides exactly into all of those numbers without a remainder. Also called GCF (Greatest Common Factor) or GCD (Greatest Common Divisor). For example, HCF(12, 18) = 6 because 6 is the largest number that divides both 12 and 18.
The Prime Factorisation Method
The prime factorisation method is the standard approach taught in GCSE Maths and is the most reliable for finding LCM and HCF of larger numbers. Here is the full worked example for LCM(12, 18) and HCF(12, 18):
Step 1: Find prime factorisations
Step 2: Find LCM (take highest power of each prime)
Step 3: Find HCF (take lowest power of shared primes)
The Relationship: LCM × HCF = Product of Two Numbers
Example: LCM(12, 18) × HCF(12, 18) = 36 × 6 = 216 = 12 × 18 = 216 ✓
This relationship is extremely useful in GCSE exams: if you know any three of the four values (a, b, LCM, HCF), you can find the fourth. Important note: this relationship holds for exactly two numbers only. For three or more numbers, there is no simple equivalent formula.
How to Find Prime Factorisation
Prime factorisation means writing a number as a product of its prime factors. There are two standard methods:
Method 1: Division Method (Ladder Method)
Divide repeatedly by the smallest prime that goes into the number:
Method 2: Factor Tree
Split the number into any two factors, then split each factor until all branches end in primes. Both methods give the same result (the fundamental theorem of arithmetic guarantees this).
LCM and Adding/Subtracting Fractions
The LCM is essential when adding or subtracting fractions with different denominators. The LCM of the denominators gives the smallest common denominator, making the calculation as simple as possible.
Example: Calculate 1/4 + 1/6
- Find LCM(4, 6) = 12
- Convert 1/4 = 3/12 (multiply top and bottom by 3)
- Convert 1/6 = 2/12 (multiply top and bottom by 2)
- Add: 3/12 + 2/12 = 5/12
Using the LCM ensures you get the smallest possible common denominator, giving answers in the simplest form from the start.
HCF and Simplifying Fractions
The HCF is used to simplify fractions to their lowest terms. Divide both the numerator and denominator by their HCF.
Example: Simplify 36/48
- Find HCF(36, 48)
- 36 = 2² × 3², 48 = 2⁴ × 3
- HCF = 2² × 3 = 12
- 36 ÷ 12 = 3; 48 ÷ 12 = 4
- Simplified: 3/4
The Euclidean Algorithm for HCF
The Euclidean algorithm is an efficient method for finding HCF that does not require prime factorisation. It uses repeated division:
HCF(48, 18) using Euclidean Algorithm
LCM and HCF for Three or More Numbers
The prime factorisation method extends naturally to three or more numbers:
- For LCM of three numbers: find all prime factorisations, then take the highest power of each prime across all three numbers.
- For HCF of three numbers: take only primes that appear in all three factorisations, using the lowest power found in any.
You can also use a step-by-step approach: LCM(a, b, c) = LCM(LCM(a, b), c). Similarly, HCF(a, b, c) = HCF(HCF(a, b), c).
GCSE Exam Technique for LCM and HCF
In GCSE Maths (Edexcel, AQA, OCR), LCM and HCF questions typically ask you to:
- Find LCM/HCF using prime factorisation: Always show your factor trees or ladder method, as marks are awarded for method, not just the answer.
- Apply to real-world contexts: “Two buses leave the station at the same time. Bus A comes every 12 minutes, Bus B every 18 minutes. When do they next arrive together?” Answer: LCM(12, 18) = 36 minutes.
- Recognise disguised questions: Questions about “largest square tile” or “largest group” are HCF questions; questions about “first time both events coincide” are LCM questions.
| Numbers | LCM | HCF | LCM × HCF | Product a×b |
|---|---|---|---|---|
| 4, 6 | 12 | 2 | 24 | 24 ✓ |
| 12, 18 | 36 | 6 | 216 | 216 ✓ |
| 15, 25 | 75 | 5 | 375 | 375 ✓ |
| 8, 12 | 24 | 4 | 96 | 96 ✓ |
| 7, 11 | 77 | 1 | 77 | 77 ✓ |
| 24, 36 | 72 | 12 | 864 | 864 ✓ |
Frequently Asked Questions
What is LCM?
LCM stands for Lowest Common Multiple (also Least Common Multiple). It is the smallest positive number that all the given numbers divide into exactly. For LCM(4, 6): multiples of 4 are 4, 8, 12, 16...; multiples of 6 are 6, 12, 18...; the smallest number in both lists is 12. LCM is used when finding a common denominator for fraction addition and subtraction, and in real-world timing problems (when do two repeating events coincide?).
What is HCF?
HCF stands for Highest Common Factor (also GCF — Greatest Common Factor, or GCD — Greatest Common Divisor). It is the largest number that divides exactly into all the given numbers. For HCF(12, 18): factors of 12 are 1, 2, 3, 4, 6, 12; factors of 18 are 1, 2, 3, 6, 9, 18; the largest number in both lists is 6. HCF is used to simplify fractions, divide groups into equal parts, and find the largest possible tile/square/segment size in practical problems.
How do you find LCM and HCF?
The prime factorisation method is the GCSE standard: (1) Find prime factorisation of each number. (2) For LCM: multiply highest powers of all primes that appear in any factorisation. (3) For HCF: multiply lowest powers of primes that appear in all factorisations. Example: LCM(12, 18): 12 = 2²×3, 18 = 2×3². LCM = 2²×3² = 36. HCF: shared primes 2 (lowest power 2¹) and 3 (lowest power 3¹). HCF = 2×3 = 6.
What is the relationship between LCM and HCF?
For any two positive integers a and b: LCM(a, b) × HCF(a, b) = a × b. Verification with 12 and 18: LCM(12,18) × HCF(12,18) = 36 × 6 = 216 = 12 × 18 = 216. This relationship is a classic GCSE exam question: “LCM is 36, HCF is 6, one number is 12 — find the other.” Solution: other number = LCM × HCF ÷ 12 = 216 ÷ 12 = 18. Remember this applies to exactly two numbers only.
How is LCM used in adding fractions?
When adding fractions with different denominators, you need a common denominator. Using the LCM gives the lowest common denominator, keeping numbers small. Example: 5/12 + 7/18. LCM(12, 18) = 36. Convert: 5/12 = 15/36 (multiply by 3) and 7/18 = 14/36 (multiply by 2). Sum = 15/36 + 14/36 = 29/36. Using LCM means you rarely need to simplify at the end because the denominator is already the smallest possible.
How do you find prime factorisation?
Divide the number by the smallest possible prime (2, then 3, then 5, then 7, etc.) and repeat until you reach 1. Example: 84. 84 ÷ 2 = 42; 42 ÷ 2 = 21; 21 ÷ 3 = 7; 7 ÷ 7 = 1. So 84 = 2² × 3 × 7. Alternatively, use a factor tree: split 84 into 4 × 21, then 4 = 2×2 and 21 = 3×7, giving the same 2² × 3 × 7. By the Fundamental Theorem of Arithmetic, every integer greater than 1 has a unique prime factorisation.