LCM & GCD Calculator

Find the Least Common Multiple (LCM) and Greatest Common Divisor (GCD/HCF) with prime factor trees and Euclidean steps.

Quick Presets:

Enter 2 or More Positive Integers (separated by commas or spaces):

Works for 2, 3, 4, or more integers up to 10,000,000. Examples: 24, 36 or 12 18 30.

Greatest Common Divisor (GCD / HCF)

The highest integer that divides all 2 numbers without remainder.

Least Common Multiple (LCM)

The smallest positive multiple shared by all 2 numbers.

Prime Factorization Decomposition

Every integer has a unique representation as a product of prime powers:

Number 124

Prime Factors:

2^3 × 3

Number 236

Prime Factors:

2^2 × 3^2

Euclidean Algorithm Division Steps (for GCD)

Repeated division remainder method discovered by Euclid of Alexandria (~300 BC).

Step	Dividend	=	Divisor × Quotient	+ Remainder
1	36	=	24 × 1	+ 12
2	24	=	12 × 2	+ 0

* The last non-zero remainder (12) is the Greatest Common Divisor (GCD).

Understanding LCM & GCD (HCF)

Greatest Common Divisor (GCD / HCF)

The GCD (also known in India and the UK as the Highest Common Factor or HCF) is the largest integer that divides two or more numbers without leaving any remainder.

GCD(24, 36) = 12 (since 12 divides 24 and 36 evenly)

Least Common Multiple (LCM)

The LCM is the smallest non-zero positive integer that is divisible by all numbers in the set. It represents the first synchronized point where their multiples meet.

LCM(24, 36) = 72 (Multiples: 24, 48, 72... and 36, 72...)

The Fundamental Product Rule

For any two positive integers a and b, the product of their LCM and GCD is always strictly equal to the product of the numbers themselves:

LCM(a, b) × GCD(a, b) = a × b

Prime Factorization Method

Break each number down into its prime building blocks. To find the GCD, multiply the lowest powers of common primes. To find the LCM, multiply the highest powers of all prime factors.

LCM & GCD Calculator FAQs

Common questions on factors, multiples, and Euclidean arithmetic.

What is the difference between HCF and GCD?

There is no difference in mathematical meaning. HCF stands for Highest Common Factor (widely used in CBSE, ICSE, and British curricula), while GCD stands for Greatest Common Divisor (standard in American textbooks). They refer to the exact same number.

Does the LCM × GCD = a × b formula work for three or more numbers?

No. The formula LCM(a, b) × GCD(a, b) = a × b strictly applies only to pairs of numbers. For three or more numbers, prime factorization or repeated Euclidean reduction must be used.

What does it mean if the GCD of two numbers is 1?

When GCD(a, b) = 1, the numbers are called 'coprime' or 'relatively prime'. They share no common divisors other than 1 (for example, 8 and 15). Their LCM is simply their product: 8 × 15 = 120.

How is LCM used in real-life problems?

LCM is fundamental whenever cyclical events repeat at different intervals: determining when traffic lights change simultaneously, calculating when two runners on a circular track lap each other, or adding fractions with unlike denominators.

Is this LCM & GCD calculator free and private?

Yes! The algorithms run 100% locally in your web browser. No data is sent to external servers, providing instantaneous results with zero latency and complete privacy.

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