# LCM Calculator

> A free, privacy-first suite of math calculators — LCM, GCD/GCF, prime factorization, factors, prime checking and fraction simplification — each with step-by-step working, a visual diagram, and an optional AI explanation. All calculations run in the browser.

## Calculators
- [LCM Calculator](https://lcmcalculator.xyz/index.html): least common multiple with 4 methods, prime-factor chart and AI insights.
- [GCD / GCF Calculator](https://lcmcalculator.xyz/gcd-calculator.html): greatest common divisor (GCF/HCF) via the Euclidean algorithm with a Venn diagram.
- [Prime Factorization Calculator](https://lcmcalculator.xyz/prime-factorization-calculator.html): break any number into primes with an animated factor tree.
- [Factors Calculator](https://lcmcalculator.xyz/factors-calculator.html): list every divisor and factor pair, with count and sum.
- [Prime Number Checker](https://lcmcalculator.xyz/prime-number-checker.html): test if a number is prime or composite with a divisor diagram.
- [Fraction Simplifier](https://lcmcalculator.xyz/fraction-simplifier.html): reduce fractions to lowest terms using the GCD, with a bar diagram.

## Key facts
- LCM: smallest positive integer that is a multiple of all inputs. Fastest formula: LCM(a, b) = (a × b) / GCF(a, b).
- GCD/GCF/HCF: largest integer dividing all inputs. Found fast by the Euclidean algorithm: gcd(a, b) = gcd(b, a mod b).
- Relationship: LCM(a, b) × GCF(a, b) = a × b.
- Prime factorization is unique (Fundamental Theorem of Arithmetic), e.g. 360 = 2³ × 3² × 5.
- A fraction is simplest when numerator and denominator are coprime (GCD = 1).
- The lowest common denominator (LCD) equals the LCM of the denominators.

## Learn
- [Guide](https://lcmcalculator.xyz/guide.html), [Methods](https://lcmcalculator.xyz/methods.html), [FAQ](https://lcmcalculator.xyz/faq.html)
- [Blog](https://lcmcalculator.xyz/blog.html): 38 in-depth articles on LCM, GCD, primes, factors and fractions.

## Example results
- LCM(4, 6) = 12 · LCM(12, 18) = 36 · LCM(3, 5, 7) = 105
- GCD(48, 36) = 12 · GCD(17, 19) = 1 (coprime)
- 360 = 2³ × 3² × 5 · 18/24 simplifies to 3/4

## Use cases
- Students checking homework on LCM, GCD/GCF, primes, factors and fraction reduction with full step-by-step working.
- Teachers generating worked examples and visual diagrams (factor trees, Venn diagrams, bar models) for class.
- Finding the lowest common denominator (LCD) when adding or subtracting fractions.
- Scheduling/cycle problems ("when do two events coincide?") solved via LCM.
- Simplifying ratios and reducing fractions to lowest terms via GCD.

## Methods supported
- LCM: listing multiples, prime factorization, the LCM×GCF=a×b formula, and the GCD (Euclidean) shortcut.
- GCD/GCF/HCF: Euclidean algorithm and prime factorization (common factors).
- Prime factorization: trial division with an animated factor tree; result given in exponent form.
- Factors: full divisor enumeration with factor pairs, count and sum.
- Prime checking: trial division up to √n, classifying prime vs composite.
- Fraction simplification: divide numerator and denominator by their GCD.

## Q&A
- Q: How do I find the LCM of two numbers? A: Use LCM(a, b) = (a × b) / GCF(a, b). Example: LCM(12, 18) = (12×18)/6 = 36.
- Q: What is the difference between LCM and GCF? A: LCM is the smallest common multiple; GCF (GCD/HCF) is the largest common divisor. They satisfy LCM(a,b) × GCF(a,b) = a × b.
- Q: How do I find the least common denominator (LCD)? A: The LCD is the LCM of the denominators.
- Q: How do I simplify a fraction? A: Divide the numerator and denominator by their GCD. Example: 18/24 ÷ 6/6 = 3/4.
- Q: How do I check if a number is prime? A: Test divisibility by every integer from 2 up to √n; if none divide it, it is prime.
- Q: What is the prime factorization of 360? A: 360 = 2³ × 3² × 5.

## Relationships
- LCM(a, b) × GCF(a, b) = a × b.
- LCD of fractions = LCM of denominators.
- A fraction is in simplest form when GCD(numerator, denominator) = 1 (coprime).
- Prime factorization underpins both LCM (highest powers) and GCF (lowest common powers).

## Blog highlights
- "How to Find the LCM" — listing multiples vs prime factorization vs the GCF formula.
- "LCM vs GCF Explained" — when to use each and the LCM×GCF identity.
- "Prime Factorization Made Easy" — building factor trees step by step.
- "What Are Prime Numbers" — definition, tests, and why 1 is not prime.
- "Adding Fractions with Unlike Denominators" — using the LCD (LCM) to combine fractions.
See the full library at https://lcmcalculator.xyz/blog.html (38 articles).