Least Common Multiple Calculator

Calculate the LCM of multiple numbers with step-by-step solutions

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Least Common Multiple Calculator

Calculate the Least Common Multiple (LCM) of two or more numbers. The LCM is the smallest positive integer that is divisible by all given numbers.

🧮 Enter Numbers

Enter up to 20 positive integers separated by commas, spaces, or new lines.

📊 Results

Least Common Multiple
4,950
LCM(330, 75, 450, 225) = 4,950
Greatest Common Divisor
15
GCD(330, 75, 450, 225) = 15

Prime Factorization Steps

330 = 2 × 3 × 5 × 11
75 = 3 × 5 × 5
450 = 2 × 3 × 3 × 5 × 5
225 = 3 × 3 × 5 × 5
LCM = 2 × 3² × 5² × 11 = 4,950
GCD = 3 × 5 = 15

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📚 Understanding the Least Common Multiple (LCM)

What is LCM?

The least common multiple (LCM) of two or more integers is the smallest positive integer that is divisible by each of the integers. It's used in fraction arithmetic and solving problems involving periodic events.

Prime Factorization Method

Break down each number into its prime factors, then multiply the highest power of each prime that appears in any factorization. This is the most systematic way to find the LCM.

LCM and GCD Relationship

For two numbers a and b: LCM(a,b) × GCD(a,b) = a × b. This relationship helps verify calculations and understand the connection between these concepts.

Methods to Find LCM

1. Brute Force Method

List out the multiples of each number and find the smallest common multiple.

Example: Find LCM(18, 26)
18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234
26: 52, 78, 104, 130, 156, 182, 208, 234
LCM(18, 26) = 234

This method can be tedious for larger numbers, so prime factorization is preferred.

2. Prime Factorization Method

Break down each number into prime factors and multiply the highest powers of all prime factors.

Example: Find LCM(21, 14, 38)
21 = 3 × 7
14 = 2 × 7
38 = 2 × 19

LCM = 2 × 3 × 7 × 19 = 798

3. Using GCD Formula

For two numbers, you can use: LCM(a,b) = (a × b) / GCD(a,b)

Example: Find LCM(15, 20)
GCD(15, 20) = 5
LCM(15, 20) = (15 × 20) / 5 = 300 / 5 = 60

Common Applications

  • Adding Fractions: Find common denominators using LCM
  • Periodic Events: When two or more events repeat at different intervals
  • Scheduling: Finding when multiple schedules align
  • Gear Ratios: Calculating when gears return to original position
  • Music: Finding common time signatures and rhythmic patterns