What is LCM (Least Common Multiple)?
The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by all of those numbers. In other words, it's the smallest number that appears in the multiplication tables of all the given numbers.
Simple Example:
Find LCM of 4 and 6:
- Multiples of 4: 4, 8, 12, 16, 20, 24...
- Multiples of 6: 6, 12, 18, 24, 30...
The smallest common multiple is 12, so LCM(4, 6) = 12
Three Methods to Calculate LCM
1. Prime Factorization Method
This is the most reliable method for finding LCM. Follow these steps:
Step 1: Find prime factors of each number
Step 2: Take the highest power of each prime
Step 3: Multiply all highest powers
2. Division Method (Ladder Method)
This method is commonly taught in schools. It's visual and easy to follow:
Find LCM of 12 and 18:
| 2 | 12 | 18 |
| 2 | 6 | 9 |
| 3 | 3 | 9 |
| 3 | 1 | 3 |
| 3 | 1 | 1 |
LCM = 2 × 2 × 3 × 3 × 3 = 36
Divide by the smallest prime that divides at least one number. Continue until all quotients are 1.
3. Listing Multiples Method
The simplest method, best for small numbers:
Find LCM of 3 and 5:
LCM(3, 5) = 15
Write out multiples until you find the first common one.
LCM Formula Using GCD
LCM(a, b) = |a × b| ÷ GCD(a, b)
Where GCD is the Greatest Common Divisor. This formula is useful for programming and quick calculations.
Real-World Applications of LCM
Scheduling Problems
If Bus A comes every 15 minutes and Bus B every 20 minutes, LCM(15, 20) = 60 tells you they arrive together every 60 minutes.
Adding Fractions
To add 1/4 + 1/6, find LCM(4, 6) = 12 as the common denominator: 3/12 + 2/12 = 5/12
Gear Ratios
In machinery, LCM helps calculate when gears with different teeth counts return to their starting position.
Music & Rhythm
Musicians use LCM to find when different rhythmic patterns (like 3-beat and 4-beat patterns) align.
Common LCM Values
| Numbers | LCM | Numbers | LCM |
|---|---|---|---|
| 2, 3 | 6 | 6, 8 | 24 |
| 3, 4 | 12 | 8, 12 | 24 |
| 4, 5 | 20 | 9, 12 | 36 |
| 4, 6 | 12 | 10, 15 | 30 |
| 5, 6 | 30 | 12, 18 | 36 |
Pro Tip for Students
If two numbers have no common factors (they are coprime), their LCM is simply their product. For example, LCM(7, 9) = 63 because 7 and 9 share no common factors other than 1.