LCM Full Form: In mathematics, the concept of LCM is crucial. The full form of LCM is “Least Common Multiple.” Let’s delve deeper into what LCM is and how to determine it.
LCM = Least Common Multiple
To grasp the idea of LCM, it is essential to understand the fundamentals of multiples. This knowledge will make it easier to find the LCM. Let’s explore the steps to find multiples, common multiples, and ultimately, the LCM.
LCM Full Form: What is LCM?
LCM stands for “Least Common Multiple.” It is a mathematical concept used to find the smallest multiple that is common to two or more numbers. To learn how to find the LCM, you first need to understand how to find multiples and common multiples.
How to Find Multiples
To find the multiples of a number, you simply multiply that number by the integers 1, 2, 3, 4, and so on.
For example, let’s find the first few multiples of 4: LCM Full Form
- 4 x 1 = 4
- 4 x 2 = 8
- 4 x 3 = 12
- 4 x 4 = 16
- 4 x 5 = 20
- 4 x 6 = 24
- 4 x 7 = 28
- 4 x 8 = 32
- 4 x 9 = 36
- 4 x 10 = 40
Thus, the multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, and 40. This method can be applied to find the multiples of any number.
Basic Rules of Multiples
- Every number is a multiple of itself.
- Every natural number is a multiple of 1.
- Every multiple of a number is greater than or equal to the number itself.
- There is no end to the multiples of a particular number.
- A number has unlimited multiples.
- We cannot find the greatest multiple.
How to Find Common Multiples
To find common multiples, first list the multiples of each number. Then, compare the lists to find the multiples that appear in both lists. These are the common multiples. LCM Full Form
Example: Finding Common Multiples of 2 and 3
Let’s list the first few multiples of 2 and 3.
- Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, …
- Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, …
The common multiples of 2 and 3 from these lists are 6, 12, 18, 24, …
Finding the LCM
After identifying the common multiples, the least common multiple (LCM) is the smallest number that appears in both lists. In our example, the LCM of 2 and 3 is 6.
By following these steps, you can determine the LCM of any set of numbers. Understanding the process of finding multiples and common multiples is key to mastering the concept of LCM. Keep practicing, and soon you’ll be able to find the LCM with ease.
How To Find Common Multiples by Prime Factorisation? – LCM Full Form
Understanding how to find common multiples is a fundamental mathematical skill, particularly useful for working with fractions, ratios, and solving equations. One of the most efficient ways to determine common multiples is through prime factorisation, which leads us to the Least Common Multiple (LCM). LCM Full Form
How to Find Prime Factorisation
Prime factorisation involves breaking down a number into its prime factors. Prime numbers are those greater than one that have no divisors other than one and themselves. There are two primary methods to determine the prime factors of a number: the Factor Tree Method and the Common Division Method. LCM Full Form
Factor Tree Method of Prime Factorisation
The factor tree method is an intuitive and visual way to break down a number into its prime factors. Here’s how you can do it:
- Start with the smallest prime factor: Begin by dividing the number by the smallest prime number that divides it evenly.
- Continue dividing: Divide the resulting quotient by the smallest prime factor and repeat this process.
- Repeat until prime: Continue until the final quotient is a prime number.
For example, to find the prime factors of 24 using the factor tree method:
- Start with 24 and divide by 2 (the smallest prime factor).
- 24 ÷ 2 = 12.
- 12 ÷ 2 = 6.
- 6 ÷ 2 = 3.
- Since 3 is a prime number, you stop here.
The prime factorisation of 24 is or 23×32^3 \times 3.
Common Division Method of Prime Factorisation
The common division method involves dividing the number by its smallest prime factor until the quotient itself is a prime number. Here’s how you can use this method: LCM Full Form
- Divide by the smallest prime factor: Start with the smallest prime factor that divides the number evenly.
- Repeat the process: Continue dividing the resulting quotient by its smallest prime factor.
- Stop when prime: Stop when the quotient is a prime number.
For instance, to find the prime factors of 24 using the common division method:
- 24 ÷ 2 = 12.
- 12 ÷ 2 = 6.
- 6 ÷ 2 = 3.
- Since 3 is a prime number, you stop here.
The prime factorisation of 24 is .
How to Find Common Multiples by Prime Factorisation
Once you have the prime factors of two or more numbers, finding their common multiples is straightforward. Here’s the process: LCM Full Form
- Prime Factorise Each Number: Determine the prime factors of each number using the factor tree or common division method.
- Identify the Highest Powers: Note the highest power of each prime factor from all the numbers.
- Multiply These Factors: Multiply these highest powers together to get the Least Common Multiple (LCM).
Example: Finding the LCM
Let’s find the LCM of 12 and 18 using their prime factors:
- Prime factors of 12 are 22×32^2 \times 3.
- Prime factors of 18 are 2×322 \times 3^2.
Combine the highest powers of all primes: . Therefore, the LCM of 12 and 18 is 36.
LCM Full Form: Least Common Multiple
The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the given numbers. It’s crucial for working with fractions, ratios, and coordinating schedules. LCM Full Form
How to Find the LCM by Common Division Method
The common division method allows you to find the LCM of several numbers at once. Here’s how it works:
- Start with the smallest prime factor: Begin by dividing the numbers by the smallest prime number.
- Apply the 50% rule: If finding the LCM of multiple numbers, at least half of them should be divisible by the prime number. Those not divisible will “fall down” unchanged.
- Repeat until all numbers are prime: Continue this process until all numbers are prime.
Example: Finding the LCM of 48, 72, and 108
Divide by 2:
- 48 ÷ 2 = 24
- 72 ÷ 2 = 36
- 108 ÷ 2 = 54
Divide by 2 again:
- 24 ÷ 2 = 12
- 36 ÷ 2 = 18
- 54 ÷ 2 = 27
Divide by 3:
- 12 ÷ 3 = 4
- 18 ÷ 3 = 6
- 27 ÷ 3 = 9
Continue dividing until all are prime.
The prime factors combined for the LCM: .
How to Find the LCM by Prime Factorisation Method
To find the LCM using prime factorisation, first prime factorise each number, then multiply the highest power of each prime factor together.
Final Thoughts on LCM
The LCM, or Least Common Multiple, is an important concept in mathematics, helping in the simplification of fractions, ratios, and scheduling problems. Understanding how to find the LCM using methods like prime factorisation and common division ensures accuracy and efficiency.
If you found this post helpful or have suggestions for improvement, please share it with your friends. LCM Full Form