Least Common Multiple (LCM)
What is the Least Common Multiple?
The Least Common Multiple (LCM) of two or more numbers is the smallest positive number that is divisible by all the given numbers without leaving a remainder.
For example: The LCM of 4 and 6 is 12, because 12 is the smallest positive number that is divisible by both 4 and 6.
Methods to Find LCM
1. Listing Multiples Method
List the multiples of each number and find the smallest common multiple.
Example: Find the LCM of 3 and 4
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Common multiples: 12, 24, ...
LCM(3, 4) = 12
2. Prime Factorization Method
Express each number as a product of prime factors, then multiply the highest powers of each prime factor.
Example: Find the LCM of 12 and 18
12 = 2² × 3
18 = 2 × 3²
Take the highest power of each prime factor: 2² × 3² = 4 × 9 = 36
LCM(12, 18) = 36
3. Using GCD (Greatest Common Divisor)
Use the formula: LCM(a, b) = (a × b) ÷ GCD(a, b)
Example: Find the LCM of 15 and 20
Find GCD(15, 20) first using the Euclidean algorithm:
20 = 15 × 1 + 5
15 = 5 × 3 + 0
So GCD(15, 20) = 5
Now, LCM(15, 20) = (15 × 20) ÷ 5 = 300 ÷ 5 = 60
LCM(15, 20) = 60
4. Division Method
Divide the numbers by common prime factors until no common factor is left.
Example: Find the LCM of 24, 36, and 40
| 2 | 24 | 36 | 40 |
| 2 | 12 | 18 | 20 |
| 2 | 6 | 9 | 10 |
| 3 | 3 | 3 | 10 |
| 1 | 1 | 10 |
Multiply all the prime factors: 2 × 2 × 2 × 3 × 10 = 8 × 3 × 10 = 24 × 10 = 240
LCM(24, 36, 40) = 360
Special Cases and Properties
1. LCM of Co-prime Numbers
If two numbers are co-prime (their GCD is 1), then their LCM is their product.
Example: 7 and 10 are co-prime since GCD(7, 10) = 1
Therefore, LCM(7, 10) = 7 × 10 = 70
2. LCM of Multiple Numbers
To find the LCM of more than two numbers, find the LCM of the first two, then find the LCM of that result and the third number, and so on.
Example: Find the LCM of 4, 6, and 8
First, find LCM(4, 6) = 12
Then, find LCM(12, 8) = 24
LCM(4, 6, 8) = 24
3. LCM of Fractions
To find the LCM of fractions, find the LCM of the numerators and the GCD of the denominators.
Example: Find the LCM of 2/3 and 5/6
LCM of numerators: LCM(2, 5) = 10
GCD of denominators: GCD(3, 6) = 3
LCM of fractions = 10/3
Real-world Applications
1. Scheduling
Finding when events with different periods will coincide.
Example: If bus A comes every 15 minutes and bus B comes every 20 minutes, how often will they arrive at the station at the same time?
We need to find the LCM of 15 and 20.
LCM(15, 20) = 60
So both buses will arrive together every 60 minutes (or 1 hour).
2. Finding Common Denominators
When adding or subtracting fractions, we need to find a common denominator.
Example: To add 2/5 + 3/8, we need to find the LCM of 5 and 8.
LCM(5, 8) = 40
So the addition becomes: (2×8)/(5×8) + (3×5)/(8×5) = 16/40 + 15/40 = 31/40
Interactive LCM Calculator
LCM Quiz
1. What is the LCM of 12 and 18?
2. If GCD(a, b) = 4 and a × b = 64, then what is the LCM of a and b?
3. The LCM of two consecutive even numbers is always:
4. If the LCM of two numbers is 48 and their GCD is 4, which of the following could be the two numbers?
5. What is the LCM of 0 and any non-zero number?
Summary
- The Least Common Multiple (LCM) is the smallest positive number that is divisible by all given numbers.
- There are multiple ways to find the LCM:
- Listing multiples method
- Prime factorization method
- Using the GCD (LCM = (a × b) ÷ GCD)
- Division method
- Special properties:
- LCM of co-prime numbers is their product
- LCM of multiple numbers can be found by finding the LCM of two numbers at a time
- LCM of fractions requires finding the LCM of numerators and GCD of denominators
- Real-world applications include scheduling problems and finding common denominators for fractions.