Least Common Multiple Calculator

Find the smallest positive integer that is a multiple of two or more integers. Perfect for adding fractions and mathematical calculations.

Calculate LCM

Enter two positive integers:

What is the Least Common Multiple?

The Least Common Multiple (LCM) is the smallest positive integer that is a multiple of two or more numbers. It's the smallest number that appears in the multiplication tables of both numbers.

Example: LCM of 4 and 6

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32...

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...

Common multiples: 12, 24, 36, 48...

Least Common Multiple: 12

LCM Formula

The LCM can be calculated using the GCD (Greatest Common Divisor):

LCM(a, b) = |a × b| / GCD(a, b)

Example Using the Formula

Find LCM(15, 20):

GCD(15, 20) = 5
LCM(15, 20) = |15 × 20| / 5 = 300 / 5 = 60

Why LCM is Important

📚 Fraction Addition

Find common denominators when adding fractions

⏰ Scheduling

Find when events occur at the same time

🔢 Number Theory

Solve problems involving multiples

🏗️ Engineering

Calculate gear ratios and periodic cycles

Fraction Addition Example

Add 1/4 and 1/6:

LCM(4, 6) = 12
1/4 = 3/12
1/6 = 2/12
3/12 + 2/12 = 5/12

Relationship with GCF

LCM and GCF are related by the formula:

GCF(a, b) × LCM(a, b) = |a × b|

Special Cases

Case Example LCM Explanation
One number divides the other LCM(4, 8) 8 Larger number is the LCM
Prime numbers LCM(5, 7) 35 Product of the primes
Same number LCM(6, 6) 6 Number itself
Coprime numbers LCM(8, 9) 72 Product of the numbers

Multiple Numbers

For more than two numbers, find LCM step by step:

LCM(a, b, c) = LCM(LCM(a, b), c)

Practical Applications

  • 📅 Calendar Planning: When do events coincide?
  • ⚙️ Machinery: Gear rotation cycles
  • 🎵 Music: Rhythm and time signatures
  • 🏃 Sports: Race intervals and timing
  • 💰 Finance: Payment schedules and cycles

Efficiency of the Algorithm

The Euclidean algorithm makes LCM calculation very efficient:

  • Time complexity: O(log min(a,b))
  • Works well even for very large numbers
  • Ancient algorithm (over 2,300 years old)
  • Fundamental in computer science and mathematics

💡 Tip: The LCM will always be greater than or equal to the larger of the two numbers, and will be a multiple of both numbers.

Common LCM Examples

Numbers LCM Common Multiples
2, 3 6 6, 12, 18, 24...
3, 4 12 12, 24, 36, 48...
4, 5 20 20, 40, 60, 80...
6, 8 24 24, 48, 72, 96...
7, 11 77 77, 154, 231, 308...