LCM Calculator — Least Common Multiple
Find the least common multiple of up to 15 numbers using prime factorization, GCD formula, or multiples method
Calculate Least Common Multiple
Please enter at least 2 positive integers.
Example Calculations
Prime Factorization Example
Problem: Find LCM of 12, 18, 24
Prime factorizations:
12 = 2² × 3¹
18 = 2¹ × 3²
24 = 2³ × 3¹
Highest powers: 2³, 3²
LCM = 2³ × 3² = 8 × 9 = 72
GCD Formula Example
Problem: Find LCM of 15 and 20
GCD calculation:
20 = 15 × 1 + 5
15 = 5 × 3 + 0
So GCD(15, 20) = 5
LCM = |15 × 20| / 5 = 300 / 5 = 60
LCM Properties
Commutative
LCM(a,b) = LCM(b,a)
Associative
LCM(a,LCM(b,c)) = LCM(LCM(a,b),c)
Relationship with GCD
LCM(a,b) × GCD(a,b) = a × b
Applications
Adding fractions with different denominators
Scheduling problems (meeting times)
Finding common periods in cyclic events
Solving Diophantine equations
Digital signal processing
Quick Tips
LCM is always ≥ the largest input number
Prime factorization is most efficient for large numbers
If numbers are coprime, LCM = their product
Use GCD formula for just two numbers
Understanding 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 all the given numbers. It's essential for solving problems involving fractions, periodic events, and scheduling.
Three Main Methods
- •Prime Factorization: Break numbers into prime factors, take highest powers
- •GCD Formula: LCM(a,b) = |a×b| / GCD(a,b)
- •List Multiples: List multiples until finding the first common one
Prime Factorization Method
This is the most efficient method for finding LCM, especially for multiple numbers or large values.
Steps:
- Find prime factorization of each number
- Identify all unique prime factors
- For each prime, take the highest power
- Multiply all highest powers together
Real-World Applications
- •Fraction Operations: Finding common denominators
- •Scheduling: When events with different periods coincide
- •Cicada Emergence: 13-year and 17-year cycles (LCM = 221 years)
LCM vs GCD
| Aspect | LCM | GCD |
|---|---|---|
| Definition | Least Common Multiple | Greatest Common Divisor |
| Size | ≥ largest input | ≤ smallest input |
| Prime Powers | Take highest | Take lowest |
Mathematical Properties
Key Formulas
LCM(a,b) × GCD(a,b) = a × b
LCM(a,b,c) = LCM(LCM(a,b), c)
If GCD(a,b) = 1, then LCM(a,b) = a × b
LCM(a,1) = a
LCM(a,0) = 0