The common multiples of two or more integers are called their common multiples, and the smallest common multiple other than 0 is called the least common multiple of these integers.
The least common multiples of the integers a and b are denoted as [a, b]. Similarly, the least common multiples of the a, b, and c are denoted as [a, b, c]. The least common multiples of the multiple integers have the same notation.
The multiples are only the smallest and not the largest, because multiples of two numbers can be infinite.
Least Common Multiple (LCM)
The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is divisible by each of the numbers. In other words, it is the smallest number that both (or all) the numbers divide into without leaving a remainder.
Key Points:
The LCM of two numbers is the smallest number that both numbers divide evenly into.
It is useful in problems involving finding common denominators, synchronization of events, and working with ratios.
Example:
Let's find the LCM of two numbers, 6 and 8.
The multiples of 6 are: 6, 12, 18, 24, 30, 36, 42, 48, ...
The multiples of 8 are: 8, 16, 24, 32, 40, 48, ...
The smallest number that appears in both lists is 24, so the LCM(6, 8) = 24.
Methods to Find LCM:
There are a few different methods to find the LCM of two or more numbers:
1. Using Prime Factorization:
Prime factorization involves breaking down each number into its prime factors, and then using the highest power of each prime factor.
Example: Find LCM of 12 and 15:
Prime factorization of 12:
12
=
2
2
×
3
1
12=2
2
×3
1
Prime factorization of 15:
15
=
3
1
×
5
1
15=3
1
×5
1
To find the LCM, take the highest powers of all the prime factors:
2
2
2
2
,
3
1
3
1
, and
5
1
5
1
LCM =
2
2
×
3
1
×
5
1
=
4
×
3
×
5
=
60
2
2
×3
1
×5
1
=4×3×5=60
Thus, LCM(12, 15) = 60.
2. Using the Relationship Between GCD and LCM:
You can calculate the LCM using the Greatest Common Divisor (GCD) with the following formula:
LCM
(
𝑎
,
𝑏
)
=
∣
𝑎
×
𝑏
∣
GCD
(
𝑎
,
𝑏
)
LCM(a,b)=
GCD(a,b)
∣a×b∣
Example: Find LCM of 12 and 15 using GCD:
GCD of 12 and 15 is 3 (because 3 is the largest number that divides both 12 and 15).
Using the formula:
LCM
(
12
,
15
)
=
∣
12
×
15
∣
3
=
180
3
=
60
LCM(12,15)=
3
∣12×15∣
=
3
180
=60
Thus, LCM(12, 15) = 60.
3. Listing Multiples Method:
For small numbers, you can list the multiples of each number and find the smallest common multiple.
Example: Find LCM of 4 and 5:
Multiples of 4: 4, 8, 12, 16, 20, 24, ...
Multiples of 5: 5, 10, 15, 20, 25, ...
The smallest common multiple is 20, so LCM(4, 5) = 20.
LCM of More Than Two Numbers:
You can extend the method to find the LCM of more than two numbers by finding the LCM of two numbers at a time.
Example: Find LCM of 4, 5, and 6:
Find LCM(4, 5) = 20 (as shown earlier).
Now find LCM(20, 6):
Multiples of 20: 20, 40, 60, 80, ...
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...
The smallest common multiple is 60, so LCM(4, 5, 6) = 60.
LCM Calculation Using Python:
You can also compute the LCM of two or more numbers using Python:
python
import math
# Function to calculate LCM of two numbers
def lcm(a, b):
return abs(a * b) // math.gcd(a, b)
# Example: LCM of 12 and 15
print(f"LCM of 12 and 15 is: {lcm(12, 15)}")
Summary of Methods:
Method Description
Prime Factorization Find the prime factors of each number and take the highest power.
Using GCD LCM = ( \frac{
Listing Multiples List the multiples of each number and find the smallest common one.
Using Python Use the math.gcd() function and compute LCM.
Conclusion:
The LCM is useful in many areas of mathematics, especially in problems involving fractions, ratios, or scheduling events. The most commonly used methods to find the LCM are prime factorization, the GCD-LCM relationship, and listing multiples.