LCM Calculator

Calculate the Least Common Multiple using multiple methods with step-by-step solutions

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Separate numbers with commas or spaces

Least Common Multiple

LCM / SCM

Greatest Common Divisor

GCD / HCF

LCM × GCD Relationship

LCM(, ) × GCD(, ) = ×

× =

Prime Factorization

LCM = Highest power of each prime:
=

Listing Multiples Method

Multiples of :
...
Multiples of :
...

✓ The smallest common multiple is

Step-by-Step Calculation

Common LCM Examples

Numbers LCM Action

LCM Properties

Basic Properties

  • • LCM(a, 1) = a
  • • LCM(a, a) = a
  • • LCM(a, b) = LCM(b, a) (commutative)
  • • LCM(a, b, c) = LCM(LCM(a, b), c)

Important Relationships

  • • LCM(a, b) × GCD(a, b) = a × b
  • • LCM(a, b) = (a × b) / GCD(a, b)
  • • If GCD(a, b) = 1, then LCM(a, b) = a × b
  • • LCM is always ≥ max(a, b)

Common Uses of LCM

Adding Fractions

Find the least common denominator when adding or subtracting fractions with different denominators

Scheduling

Determine when events with different cycles will occur simultaneously

Music & Patterns

Find when rhythmic patterns with different beat lengths align

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About LCM Calculator

What is an LCM Calculator?

An LCM (Least Common Multiple) Calculator helps you find the smallest positive integer that is divisible by two or more numbers without leaving a remainder. The LCM is also known as the Lowest Common Multiple or Smallest Common Multiple (SCM).

How to Use This Calculator

  1. Two Numbers Mode: Enter two numbers to find their LCM
  2. Multiple Numbers Mode: Enter multiple numbers separated by commas to find the LCM of all
  3. Step-by-Step Mode: See the calculation method applied step by step
  4. Compare with GCD: View both LCM and GCD for the given numbers

Understanding LCM

What is the Least Common Multiple?

The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is perfectly divisible by each of the integers.

Example: LCM(4, 6) = 12 because:

  • 12 ÷ 4 = 3 (no remainder)
  • 12 ÷ 6 = 2 (no remainder)
  • No smaller positive integer is divisible by both 4 and 6

LCM Calculation Methods

1. Using GCD Formula

The most efficient method uses the relationship between LCM and GCD:

LCM(a, b) = (a × b) / GCD(a, b)

2. Prime Factorization Method

  1. Find the prime factorization of each number
  2. Take the highest power of each prime factor
  3. Multiply all the highest powers together

Example: LCM(12, 18)

  • 12 = 2² × 3¹
  • 18 = 2¹ × 3²
  • LCM = 2² × 3² = 4 × 9 = 36

3. Listing Multiples Method

  1. List multiples of each number
  2. Find the smallest common multiple

Example: LCM(4, 6)

  • Multiples of 4: 4, 8, 12, 16, 20, 24...
  • Multiples of 6: 6, 12, 18, 24...
  • LCM = 12

4. Division Method (Ladder Method)

  1. Write numbers in a row
  2. Divide by the smallest prime that divides at least one number
  3. Continue until all results are 1
  4. Multiply all divisors

LCM Properties

  • LCM(a, 1) = a
  • LCM(a, a) = a
  • LCM(a, b) = LCM(b, a) (commutative)
  • LCM(a, b) × GCD(a, b) = a × b
  • LCM(a, b, c) = LCM(LCM(a, b), c)

Frequently Asked Questions

What is the difference between LCM and GCD?

LCM finds the smallest number that all given numbers divide into, while GCD (Greatest Common Divisor) finds the largest number that divides all given numbers.

When do I need to find LCM?

LCM is commonly used when:

  • Adding or subtracting fractions with different denominators
  • Finding common schedules (e.g., when will two events occur at the same time again?)
  • Solving problems involving cycles or repeating patterns

How do I find the LCM of more than two numbers?

To find LCM(a, b, c), first find LCM(a, b), then find LCM(result, c). Continue this process for additional numbers.

Tip: The relationship LCM(a, b) × GCD(a, b) = a × b is very useful for quickly calculating LCM if you already know the GCD.