LCM Formulas
Master the Least Common Multiple with comprehensive formulas and methods. From basic techniques to advanced algorithms for finding LCM efficiently.
1LCM Definition
The Least Common Multiple (LCM) of two or more integers is the smallest positive integer that is divisible by all the given numbers.
Properties:
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Example:
Because 36 is the smallest number that both 12 and 18 divide into evenly.
2Fundamental GCD-LCM Relationship
The most important formula for calculating LCM uses the relationship with GCD.
Equivalently:
Example: LCM(48, 18)
Step 1: Find GCD(48, 18) = 6
Step 2: Apply formula:
Answer: 144
3Prime Factorization Method
Find LCM by taking the highest power of each prime factor that appears in either number.
Example: LCM(60, 48)
Prime factorizations:
LCM calculation:
4Division Method (Ladder Method)
A systematic approach using repeated division by prime factors.
Algorithm Steps:
1. Write the numbers in a row
2. Find the smallest prime that divides at least one number
3. Divide and write quotients below
4. Repeat until all quotients are 1
5. LCM = product of all divisors used
Example: LCM(12, 16, 20)
| Divisor | 12 | 16 | 20 |
|---|---|---|---|
| 2 | 6 | 8 | 10 |
| 2 | 3 | 4 | 5 |
| 2 | 3 | 2 | 5 |
| 2 | 3 | 1 | 5 |
| 3 | 1 | 1 | 5 |
| 5 | 1 | 1 | 1 |
5Listing Multiples Method
A simple method suitable for small numbers - list multiples until you find a common one.
Example: LCM(6, 8)
Multiples of 6:
6 × 1 = 6
6 × 2 = 12
6 × 3 = 18
6 × 4 = 24 ✓
6 × 5 = 30
Multiples of 8:
8 × 1 = 8
8 × 2 = 16
8 × 3 = 24 ✓
8 × 4 = 32
8 × 5 = 40
First common multiple: LCM(6, 8) = 24
6LCM of Multiple Numbers
Associative Property:
Example:
7Special Cases and Properties
Common Special Cases
Coprime Numbers
When GCD(a,b) = 1:
One Divides Another
When a divides b:
Same Numbers
Powers of Same Base
For :
8Mathematical Applications
🔢 Adding Fractions
To add :
📊 Periodic Events
Events with periods a and b will coincide every LCM(a,b) time units.
Example: Buses every 12 and 18 minutes meet every LCM(12,18) = 36 minutes
🧮 Number Theory
• Diophantine equations
• Modular arithmetic
• Chinese Remainder Theorem
⚙️ Engineering
• Gear ratios and timing
• Signal processing periods
• Scheduling optimization
Quick Reference
Common LCM Values
lcm(6, 8)=24
lcm(12, 18)=36
lcm(15, 20)=60
lcm(9, 12)=36
Key Relationships
GCD-LCM Formula
Divisibility
LCM(a,b) is divisible by both a and b
Bounds