LCM Calculator
Calculate the Least Common Multiple (LCM) and GCD for 2 to 10 numbers with interactive leapfrog number line race, prime factor Venn diagram, division ladder, and step-by-step methods.
Smallest positive whole number that all 3 numbers divide into evenly with zero remainder.
Watch leapfrog multiples align or explore prime factor Venn diagrams.
For 3 numbers, the direct relationship generalizes across the set:
Calculation Method Breakdown
Explore 4 different pedagogical techniques used in mathematics classrooms.
Step 1: Find the prime factorization of each number in exponential form:
Step 2: Take the highest exponent for every prime factor that appears in any number:
Itemized Calculation Summary
Ready for printing, homework submission, or quick sharing.
======================================== LCM & GCD CALCULATION SUMMARY OctaLone Tools (octalone.com/tools/lcm-calculator) ======================================== Input Numbers: 12, 18, 24 PRIMARY RESULTS: • Least Common Multiple (LCM): 72 • Greatest Common Divisor (GCD/GCF): 6 PRIME FACTORIZATION BREAKDOWN: • 12 = 2^2 × 3 • 18 = 2 × 3^2 • 24 = 2^3 × 3 ========================================
Did You Know? Math Insights
Euclid's Elements (c. 300 BC)
Proposition 34 of Book VII in Euclid's Elements proved the Least Common Multiple using the Euclidean algorithm for GCD over 2,300 years ago!
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Frequently Asked Questions
The Least Common Multiple (LCM), also known as the Lowest Common Multiple or Smallest Common Multiple, is the smallest positive integer that is evenly divisible by all numbers in a given set without leaving a remainder.
LCM is the smallest positive number that is a multiple of all given numbers (equal to or larger than the greatest input number). GCD (Greatest Common Divisor) or GCF (Greatest Common Factor) is the largest integer that divides all the given numbers evenly (equal to or smaller than the smallest input number).
First, write the prime factorization of each number in exponential form (e.g. 12 = 2² × 3¹ and 18 = 2¹ × 3²). Next, take the highest power of every prime factor that appears in any of the factorizations (2² and 3²). Finally, multiply these maximum prime powers together: 2² × 3² = 4 × 9 = 36.
Arrange the numbers in a horizontal row. Divide them by common prime divisors, bringing down numbers that are not divisible. Continue until no two numbers share a common prime factor. Multiply all the prime divisors along the left column and the remaining bottom row quotients to obtain the LCM.
For any two positive integers a and b, the product of their LCM and GCD equals the product of the two numbers: LCM(a, b) × GCD(a, b) = a × b. For example, for 12 and 18, LCM(12, 18) = 36 and GCD(12, 18) = 6; 36 × 6 = 216 = 12 × 18.
Yes! Our calculator supports between 2 and 10 positive integers simultaneously. It computes the multi-number LCM using associative pairwise Euclidean reduction and displays step-by-step methods for all numbers.
Yes, 100%. All arithmetic, prime factorization, and interactive vector visualizations run entirely in your local browser using client-side JavaScript. No numbers are ever sent to any remote server.