What is the difference between HCF and GCD?

LCM & HCF (GCD) Formula

Mathematics & Education Formula • Published on September 01, 2026 • Last updated September 01, 2026

Mathematical Equation

$$\text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b \quad \implies \quad \text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}$$

Variable Definitions

LCM(a, b)

Least Common Multiple of positive integers a and b

HCF(a, b)

Highest Common Factor (Greatest Common Divisor) of a and b

a, b

Given positive integers

Detailed Explanation

In-Depth Guide

The Least Common Multiple (LCM) is the smallest positive integer divisible by all numbers in a set, while the Highest Common Factor (HCF or GCD) is the largest integer that divides all numbers without a remainder. For any two positive integers, the product of their LCM and HCF equals the product of the numbers.

How to Calculate: Step-by-Step

1. Express each number as a product of prime factors. 2. HCF: Multiply the lowest powers of common prime factors. 3. LCM: Multiply the highest powers of all prime factors. 4. Alternatively, use Euclidean algorithm to find HCF, then compute LCM = (a × b) / HCF.

Worked Calculation Example

Numbers: 12 and 18 - Prime factorizations: 12 = 2^2 × 3, 18 = 2 × 3^2 - HCF(12, 18) = 2^1 × 3^1 = 6 - LCM(12, 18) = 2^2 × 3^2 = 36 - Verification: 36 × 6 = 216 = 12 × 18.

Common Use Cases

Fraction addition & common denominator calculation
Periodic event & signal synchronization
Equal grouping & package partitioning

Frequently Asked Questions

HCF (Highest Common Factor) and GCD (Greatest Common Divisor) are identical terms for the same mathematical concept.

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