Calcia Logo
Calcia
Calculate
Pure & Applied MathematicsEntry № 03.06Documented Formula

LCM & HCF (GCD) Calculator with Step-by-Step Euclidean Algorithm

Determines the Least Common Multiple (LCM) and Highest Common Factor (HCF / GCD) of two or more integers via the Euclidean division algorithm.

Popular Searches:Find LCM and HCF of 24 and 36 step by stepEuclidean algorithm for GCD formulaRelation between LCM and HCF: LCM × HCF = a × b
Euclidean Algorithm LCM & HCF (GCD) Engine
LCM(a,b) × HCF(a,b) = a × b
Euclidean Division Steps
36 = 24 × 1 + 12
24 = 12 × 2 + 0
Highest Common Factor (HCF / GCD)
12
Least Common Multiple (LCM)
72

Identity Proof: 24 × 36 = 864 = HCF (12) × LCM (72)

Mathematical Proof & FormulaStandard Mathematical Notation

How This Calculation Formula is Formulated

Calculates the Greatest Common Divisor (GCD/HCF) via successive Euclidean division, and derives the Least Common Multiple (LCM) through the fundamental relation a · b = LCM(a, b) · GCD(a, b).

Standard Algebraic Representation
GCD(a, b) via a mod b ; LCM(a, b) = (|a · b|) / GCD(a, b)
Documented algebraic formula with transparent derivation and reference notes.

Variable Definitions & Measurement Units

a, b
Integers
Two non-zero integers evaluated.
GCD
Greatest Common Divisor
Largest positive integer dividing both a and b.
LCM
Least Common Multiple
Smallest positive integer divisible by both a and b.
Step-by-Step Calculation Example

LCM & HCF of 24 and 36

Finding the greatest common divisor and least common multiple of 24 and 36.

1
Step 1: 36 mod 24
36 = 24 × 1 + 12 (Remainder 12)
Euclidean step 1
2
Step 2: 24 mod 12
24 = 12 × 2 + 0 (Remainder 0)
Euclidean step 2 — termination
3
HCF Identification
HCF(24, 36) = 12
Last non-zero remainder
4
LCM Computation
LCM = (24 × 36) / 12 = 864 / 12 = 72
Product divided by HCF
Conclusion: For integers 24 and 36, the Highest Common Factor (HCF) is 12 and the Least Common Multiple (LCM) is 72.
In-Depth Editorial Analysis

The Computational Efficiency of the Euclidean Algorithm

Described by Euclid in Book VII of his Elements around 300 BC, the Euclidean division algorithm is one of the oldest numerical algorithms still in widespread practical use.

Its worst-case time complexity follows Lamé's Theorem: the number of division steps never exceeds five times the number of digits in the smaller integer (O(log(min(a, b)))), vastly outperforming brute-force prime factorization for massive cryptographic numbers.

Frequently Asked Questions

Frequently Asked Questions About LCM & HCF (GCD) Calculator with Step-by-Step Euclidean Algorithm

What is the relation between LCM and HCF?

For any two positive integers a and b, their product equals the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b).

How do you find LCM of three numbers?

Find the LCM of the first two numbers, then find the LCM of that result with the third number: LCM(a, b, c) = LCM(LCM(a, b), c).

Related Online Calculators

Frequently referenced alongside LCM & HCF (GCD) Calculator with Step-by-Step Euclidean Algorithm.

View All Pure & Applied Mathematics →
Proportion№ 03.09

Ratio Simplifier & Equivalent Proportions Calculator

Reduces ratios a : b to lowest fractional terms by dividing by their greatest common divisor and computes scaled equivalent proportions.

Simplified = [a / GCD(a,b)] : [b / GCD(a,b)]Open Tool
Number Theory№ 03.10

Prime Factorization & Factor Tree Calculator

Decomposes any positive integer into its canonical prime factor product with exponential powers, listing all divisors and primality verification.

n = p₁ᵃ¹ · p₂ᵃ² · ... · pₖᵃᵏ (Fundamental Theorem of Arithmetic)Open Tool
Proportion№ 03.01

Percentage Calculator

Evaluates three primary proportional cases: percentage of a number, percentage difference, and percentage change.

Δ% = ((x₂ - x₁) / |x₁|) × 100Open Tool

Deterministic Precision & 100% Client-Side Privacy

All calculations execute in your local browser using IEEE 754 double-precision floating-point arithmetic. Your figures and financial metrics remain private and are never uploaded or saved to any cloud servers.