M.RajaRao, MSc, MEd. S.A(Mathematics)
Mathematics Fundamentals
Least Common Multiple (LCM)
The smallest number that is a multiple of two or more numbers. Essential for fractions, scheduling, and solving real-world synchronization problems.
What is LCM?
The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers.
In simpler terms, it's the smallest number that can be divided evenly by each of the numbers without leaving a remainder.
Example: LCM of 6 and 9
Multiples of 6: 6, 12, 18, 24...
Multiples of 9: 9, 18, 27...
Visualizing LCM(4, 6)
Both bars align perfectly at length 12.
How to Find the LCM
There are three primary methods to calculate the Least Common Multiple. Choose the one that works best for your numbers.
1. Listing Multiples
List the multiples of each number and find the smallest one that appears in all lists.
Example: LCM of 3, 4, 5
- 3: 3, 6... 57, 60
- 4: 4, 8... 56, 60
- 5: 5, 10... 55, 60
LCM = 60
2. Prime Factorization
Find prime factors using the Ladder Method for each number, then take the highest power of each prime.
Example: 12, 16, 18
| 2 | 12 |
| 2 | 6 |
| 3 | 3 |
| 1 |
| 2 | 16 |
| 2 | 8 |
| 2 | 4 |
| 2 | 2 |
| 1 |
| 2 | 18 |
| 3 | 9 |
| 3 | 3 |
| 1 |
3. Division Method
Divide all numbers by common primes until no more division is possible. This is often called the Ladder Method.
Example: LCM of 12, 15
| 3 | 12, 15 |
| 2 | 4, 5 |
| . | 2, 5 |
Multiply side & bottom:
3 × 2 × 2 × 5 = 60
Real-Life Applications
Scheduling Events
If Bus A arrives every 2 hours and Bus B every 3 hours, finding LCM(2,3)=6 tells you they arrive together every 6 hours.
Synchronizing Cycles
Two flashing lights with intervals of 6s and 8s will flash simultaneously every LCM(6,8) = 24 seconds.
Working with Fractions
To add fractions like:
You need the common denominator. LCM(4,6) = 12.
Tiling & Patterns
Tiles of 4 inches and 6 inches will align perfectly at every LCM(4,6) = 12 inches boundary.
Gears & Mechanics
A gear with 5 teeth and one with 8 teeth will return to the starting position after LCM(5,8) = 40 teeth movements.
Musical Beats
Poly-rhythms repeating every 3 and 4 beats align at measure 12 (LCM of 3 and 4).
Instant LCM Calculator
Enter numbers separated by commas to find their Least Common Multiple and see the steps.
Answer
LCM of 12, 15, 20 is 60
1 Method 1: Listing Multiples
2 Method 2: Prime Factorization (Ladders)
3 Method 3: Division Method (Ladder)
Test Your Knowledge
Can you score 100% on this LCM Quiz?
LCM Practice Quiz
10 questions to master your understanding of Least Common Multiples.
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