What is the Greatest Common Divisor (GCD)?

The GCD (also called Greatest Common Factor or GCF) is the largest positive integer that divides two or more numbers without leaving a remainder.

What is the Least Common Multiple (LCM)?

The LCM is the smallest positive integer that is divisible by all given numbers. For two numbers a and b, LCM(a, b) = |a * b| / GCD(a, b).

How does the Euclidean algorithm find the GCD?

The Euclidean algorithm repeatedly computes the remainder of dividing the larger number by the smaller number until the remainder is zero. The last non-zero divisor is the GCD.

Can I calculate GCD and LCM for more than two numbers?

Yes, enter any list of positive integers separated by commas or spaces.

GCD & LCM Calculator - Greatest Common Divisor & Least Common Multiple

What is GCD and LCM in Number Theory?

The Greatest Common Divisor (GCD), also widely termed the Greatest Common Factor (GCF), is the highest integer that divides each number in a set evenly without a remainder. Conversely, the Least Common Multiple (LCM) is the smallest positive number that is a multiple of every number in the set. Together, they are essential for simplifying fractions, finding common denominators, and cryptography key generation.

Key Capabilities

Multi-Number Analysis: Computes GCD and LCM across two, three, or more integers simultaneously.
Euclidean Algorithm Steps: Illustrates classical division remainder progressions step-by-step.
Instant Fraction Simplifier Aid: Quickly identify common factors to reduce complex algebraic fractions.
100% In-Browser Execution: Zero server latency, instant updates, and privacy-first calculation.

How to Calculate GCD and LCM

Type two or more numbers separated by commas or spaces (e.g. 24, 36, 60).
The tool instantly outputs the highest common factor (GCD) and lowest common multiple (LCM).
Read the Euclidean algorithm division steps to verify homework or algebraic proofs.

Frequently Asked Questions

What does it mean if the GCD is 1?

If $GCD(a, b) = 1$, the numbers are classified as coprime or relatively prime (such as 8 and 9). They share no common factors other than 1.

What is the relationship between GCD and LCM?

For any two positive numbers $a$ and $b$, their product equals the product of their GCD and LCM: $a imes b = GCD(a, b) imes LCM(a, b)$.

Can I enter negative numbers?

By standard convention, GCD and LCM are positive integers. The calculator automatically converts negative inputs to absolute positive values.

Is this tool free?

Yes, it is 100% free forever.

GCD & LCM Calculator

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Euclidean Division & Factorization Steps: