Different ways to find LCM: You may have come across LCM in your mathematical endeavors most of the time. But, wait a second, have you ever paused and thought about the history of LCM? Have you ever felt the spark of curiosity about the founders of LCM or when they found it and why they found it? In this blog along with methods to find LCM, you will also get to delve deeper into the history of LCM.
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History:
Today we can see the theory of LCM plays an important role in different sections of mathematics, such as number Theory and Algebra. For the first time, LCM was invented around 200 or 300 B.C. It was Euclid, a Greek mathematician, who made notable contributions to the LCM. Indian mathematicians such as Brahmagupta and Diophantus also made important progress in the field of LCM.
Definition of LCM:
The abbreviation of LCM is Least Common Multiple. It is the least common multiple of two or more numbers and is the smallest positive integer that is a multiple of all those numbers. It’s just like finding the lowest common denominator for a set of fractions.
The general and most formal way of defining LCM is by considering a set of numbers where
S = {a1, a2, a3, … .. an}, the LCM(S) is the smallest positive integer that satisfies the following condition:
For example, the LCM of 2 and 3 is 6 because 6 is the smallest number that is divisible by both 2 and 3.
3 Different Methods and ways to find LCM
1. Prime Factorization Method to find LCM
In this method, we break down the given number into its prime factors which are divisible by itself or 1.
Steps to find:
Find the prime factorization of each number in the given set.
Point out the highest power of each prime factor that appears in any of the factorizations.
Multiply the prime factors with the highest power.
Example:
Find the LCM of 12 and 18
Prime factors of the number 12 are 2, 2, 3 (that is 2x2x3)
Prime factors of the number 18 are 2, 2, 3 (that is 2x3x3)
Therefore, LCM is 2x2x3x3 = 4×9 = 36
Application: This method is efficient for finding LCMs of numbers with Prime factors. It is useful, particularly when dealing with big numbers.
2. Listing multiples to find LCM
Listing multiplies the way of listing the multiples of each number in a set until you find a common multiple. Once you find it keep checking for smaller multiples until you find the least one.
Example:
Find the LCM of 6 and 8.
List the multiples of the given numbers
3 ->3, 6, 9, 12, 15, 18, 21 …
And 7 -> 7, 14, 21, 28, 35, 42, …
Therefore, the least common multiple is 21.
Application: This method is applicable for the sets with lesser numbers. However, it can become a complicated process for larger sets.
3. Division Method to find LCM
This method is more suitable for the repeated division of prime numbers this method is also called as greatest common divisor method.
Steps to find:
Note down the given number from the set.
Now start dividing the given number with the smallest prime number that divides any of them.
If you are unable to divide the given number by prime number then bring it down as it is.
Start repeating step 2 with the obtained quotients in the previous step, ignore if 1 is present.
Continue dividing by the prime number until all the numbers become 1.
Write down the LCM which is a product of all the prime numbers you have encountered in the division process.
Example:
Find LCM of 36, 48, 60
Solution:
Note down – 36, 48, 60
Divide by using a small prime factor –
Divide the number 36 by 2: 36 / 2 = 18
Divide the number 48 by 2: 48 / 2 = 24
Divide the number 60 by 2: 60 / 2 = 30
Repeat the second step by using the obtained quotients.
Divide the number 18 by 2: 18 / 2 = 9
Divide the number 24 by 2: 24 / 2 = 12
Divide the number 30 by 2: 30 / 2 = 15
Continue until all numbers become 1
Divide the number 9 by 3: 9 /3 = 3
Divide the number 12 by 2: 12 / 2 = 6
Divide the number 15 by 3: 15 / 3 = 5
Write down the prime numbers you encountered
GCD (36, 48, 60) = 2 x 2 x 3 = 12.
Application:
LCM is used in time and motion, Scheduling planning and also fraction operations.
This article has provided you with a brief history and different methods to calculate the LCM. Further with real-time applications.
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