The Highest Common Factor (HCF), also called Greatest Common Divisor (GCD), is the largest positive integer that divides each of the given numbers without a remainder. For example, HCF(12, 18) = 6.

What is the Euclidean algorithm?

The Euclidean algorithm finds the GCD by repeatedly dividing the larger number by the smaller and taking the remainder, until the remainder is 0. The last non-zero remainder is the GCD.

HCF Calculator — GCD of 2 or More Numbers

Enter Numbers

Your Results

HCF (GCD) 12

Method HCF(24, 36) = 12

HCF (GCD) 12

LCM 72

Product of Numbers 864

Coprime? No

Count 2

Prime Factorization

24 = 2³ × 3

36 = 2² × 3²

HCF = 2² × 3 = 12

What Is the HCF / GCD Calculator?

The HCF Calculator (also called a GCD Calculator) is a free online tool that finds the Highest Common Factor — or Greatest Common Divisor — of two or more positive integers. The HCF of a set of numbers is the largest positive integer that divides every number in the set without leaving a remainder. For instance, HCF(12, 18) = 6 because 6 is the biggest whole number that goes evenly into both 12 and 18.

Whether you need to simplify fractions, solve number-theory homework, or determine the largest square tile for a rectangular floor, knowing how to find the HCF of two numbers quickly is an essential maths skill.

Key Features

Unlimited inputs: Calculate the HCF of 2, 3, or more numbers in a single operation.
Instant results: HCF, LCM, product of numbers, and coprime status update in real time as you type.
Prime factorization breakdown: See each number decomposed into its prime factors and the common-factor extraction step by step.
Coprime detection: Instantly know whether the entered numbers are coprime (HCF = 1).
LCM included: The companion Least Common Multiple is calculated automatically using the HCF–LCM relationship.
Mobile-friendly & accessible: Fully responsive layout with keyboard navigation and ARIA labels.

Formulas — How to Find HCF of Two Numbers

1. Prime Factorization Method

Express each number as a product of prime powers. Then take the lowest power of every prime that appears in all factorizations and multiply them together.

HCF = ∏ p^{min(e₁, e₂, …)}

2. Euclidean Algorithm (Division Method)

Divide the larger number by the smaller number and note the remainder. Replace the larger number with the smaller and the smaller with the remainder. Repeat until the remainder is 0. The last non-zero remainder is the HCF.

GCD(a, b) = GCD(b, a mod b), with GCD(n, 0) = n

3. Listing-Factors Method

List every factor of each number, then pick the largest factor that appears in every list.

4. HCF–LCM Relationship

For two numbers: HCF(a, b) × LCM(a, b) = a × b

How to Use This Calculator — Step by Step

Enter your numbers: Type two or more positive integers into the text area, separated by commas or spaces (e.g., 24, 36, 48).
Click “Calculate” or simply watch the results update instantly as you type.
Read the results: The HCF, LCM, product, coprime status, and input count appear in the results card.
Review the steps: Scroll to the Prime Factorization section to see how each number was decomposed and how the common factors were extracted.

Practical Examples — HCF Calculation Methods Explained

Example 1 — Two numbers (Prime Factorization)

Find HCF(24, 36).
24 = 2³ × 3 | 36 = 2² × 3²
Common primes at lowest powers: 2² × 3 = 12

Example 2 — Two numbers (Euclidean Algorithm)

Find HCF(48, 18).
48 ÷ 18 = 2 remainder 12
18 ÷ 12 = 1 remainder 6
12 ÷ 6 = 2 remainder 0 → HCF = 6

Example 3 — Three numbers

Find HCF(48, 60, 72).
48 = 2⁴ × 3 | 60 = 2² × 3 × 5 | 72 = 2³ × 3²
Common primes at lowest powers: 2² × 3 = 12

Real-World Use Cases

Simplifying fractions: Divide both the numerator and denominator by their HCF to reduce a fraction to its simplest form (e.g., 24/36 ÷ 12 = 2/3).
Equal distribution: Split items into the largest possible equal groups — for example, distributing 48 apples and 60 oranges into bags with the same mix.
Tiling & flooring: Determine the largest square tile size that covers a rectangular room with no cutting (e.g., a 48 cm × 60 cm area uses 12 cm tiles).
Music theory: Finding rhythmic common denominators when layering time signatures.
Cryptography: The Euclidean algorithm is a building block of RSA key generation.

Understanding Your Results

HCF (GCD): The largest integer that divides all entered numbers evenly.
LCM: The smallest positive integer that is a multiple of every entered number. Calculated via HCF(a, b) × LCM(a, b) = a × b.
Product of Numbers: Simply all entered values multiplied together — useful for verifying the HCF–LCM relationship.
Coprime? Displays Yes when HCF = 1 (the numbers share no common factor other than 1) and No otherwise.
Prime Factorization steps: Shows each number expressed as a product of prime powers, followed by the common-factor extraction line.

Tips & Best Practices

For very large numbers, the Euclidean algorithm is far more efficient than listing all factors.
When working with three or more numbers, compute HCF pair-wise: HCF(a, b, c) = HCF(HCF(a, b), c).
If you only need the LCM, use the formula LCM = (a × b) / HCF(a, b) rather than listing multiples.
Enter numbers in any order — the HCF is the same regardless of sequence.
Use the coprime check to quickly verify whether two numbers share a factor before simplifying a fraction.

Common Mistakes to Avoid

Confusing HCF with LCM: HCF is the largest common divisor; LCM is the smallest common multiple. They serve opposite purposes.
Forgetting to take the lowest power: In the prime-factorization method, always pick the minimum exponent of each common prime, not the maximum.
Stopping the Euclidean algorithm too early: Keep dividing until the remainder is exactly 0; the last non-zero remainder is the answer.
Including 0 or negative numbers: HCF is defined for positive integers. Zero is divisible by every integer, so HCF(0, n) = n by convention.
Assuming coprime means prime: Two composite numbers can still be coprime (e.g., HCF(8, 15) = 1).

Frequently Asked Questions

Q: What is the difference between HCF and GCD?

They are the same concept. “Highest Common Factor” (HCF) is used primarily in British and Commonwealth curricula, while “Greatest Common Divisor” (GCD) is more common in American textbooks.

Q: Can the HCF be larger than the smallest input number?

No. The HCF can be at most equal to the smallest number in the set (which happens when the smallest number divides all the others).

Q: What is the relationship between HCF and LCM?

For any two positive integers a and b: HCF(a, b) × LCM(a, b) = a × b. This identity lets you compute one from the other.

Q: How do I find the HCF of fractions?

HCF of fractions = HCF(numerators) ÷ LCM(denominators). For example, HCF(2/3, 4/5) = HCF(2, 4) / LCM(3, 5) = 2/15.

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