# LCM Calculator — Full AI Context (llms-full.txt)

> Extended, page-by-page knowledge base for lcmcalculator.xyz. A free, privacy-first suite of number-theory calculators. All computation runs client-side in the browser; no data is sent to a server. Each tool shows step-by-step working, a visual diagram, and an optional AI explanation.

Site: https://lcmcalculator.xyz
Last updated: 2026-06-22

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## 1. LCM Calculator  (/index.html)
Finds the Least Common Multiple of two or more integers.
Definition: the LCM of a set of integers is the smallest positive integer that is a multiple of every number in the set.
Methods shown: (a) listing multiples, (b) prime factorization (take the highest power of each prime), (c) the identity LCM(a,b) = (a×b) / GCF(a,b).
Worked example: LCM(12, 18) — 12 = 2²×3, 18 = 2×3²; highest powers 2²×3² = 36. Check: 12×18 / GCF(12,18)=216/6=36.
Common uses: least common denominator (LCD) when adding fractions, repeating-cycle/scheduling problems.
Examples: LCM(4,6)=12 · LCM(3,5,7)=105 · LCM(8,12)=24.

## 2. GCD / GCF Calculator  (/gcd-calculator.html)
Finds the Greatest Common Divisor (also GCF / HCF) of two or more integers.
Definition: the largest positive integer that divides every number in the set without a remainder.
Methods shown: Euclidean algorithm (repeated remainder) and prime factorization (product of common primes at lowest powers).
Worked example: GCD(48, 36) — 48=2⁴×3, 36=2²×3²; common 2²×3 = 12. Euclid: 48 mod 36 = 12, 36 mod 12 = 0 → 12.
Common uses: simplifying fractions/ratios. GCD(17,19)=1 means coprime.

## 3. Prime Factorization Calculator  (/prime-factorization-calculator.html)
Expresses a number as a product of primes, with an animated factor tree.
Method: trial division by successive primes; output in exponent form.
Worked example: 360 = 2³ × 3² × 5.
Common uses: foundation for computing LCM (highest powers) and GCF (lowest common powers).

## 4. Factors Calculator  (/factors-calculator.html)
Lists every divisor of a number, plus factor pairs, the count of factors and their sum.
Worked example: factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24 (8 factors); pairs (1,24)(2,12)(3,8)(4,6).
Common uses: finding common factors, understanding divisibility.

## 5. Prime Number Checker  (/prime-number-checker.html)
Determines whether a number is prime or composite.
Method: trial division by integers from 2 up to √n. Note 1 is neither prime nor composite; 2 is the only even prime.
Worked example: 97 is prime (no divisor ≤ 9); 91 = 7×13 is composite.

## 6. Fraction Simplifier  (/fraction-simplifier.html)
Reduces a fraction to lowest terms and handles improper fractions / mixed numbers.
Method: divide numerator and denominator by their GCD; a fraction is simplest when GCD = 1.
Worked example: 18/24 — GCD=6 → 3/4. 45/10 → 9/2 = 4½.

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## Key relationships (canonical facts)
- LCM(a, b) × GCF(a, b) = a × b
- Least Common Denominator (LCD) = LCM of the denominators
- A fraction is in simplest form ⇔ GCD(numerator, denominator) = 1
- Prime factorization is the basis for both LCM (highest prime powers) and GCF (lowest common prime powers)

## Frequently asked questions
- How to find LCM of two numbers? LCM(a,b) = (a×b)/GCF(a,b). e.g. LCM(12,18)=36.
- LCM vs GCF? LCM = smallest common multiple; GCF = largest common divisor.
- How to find the LCD for fractions? Take the LCM of the denominators.
- How to simplify a fraction? Divide top and bottom by their GCD.
- How to check if a number is prime? Test divisibility up to √n.
- Is 1 prime? No — by definition 1 is neither prime nor composite.

## Blog (https://lcmcalculator.xyz/blog.html)
A library of ~38 tutorials covering LCM methods, GCF, prime factorization, prime numbers, factor pairs, adding/subtracting fractions with unlike denominators, ratios, and number-theory fundamentals — written for students and teachers.

## About
lcmcalculator.xyz is free, requires no sign-up, runs entirely in the browser (works offline as a PWA), and is designed to teach the method, not just give the answer.