How to Find LCM Using the Long Division Method with Examples
How Do We Find Out the LCM of a Number?
LCM by division is the process of finding the least common multiple of a given group of integers by dividing them by a common prime number. For this, we need to completely understand the prime factors of the given numbers. Prime factorization and listing the multiples are two additional techniques for calculating LCM.
LCM by Division Method
The LCM by division technique finds the LCM (Least Common Multiple) of integers by reducing them by shared prime numbers. This technique of determining the LCM of numbers is one of the standards and provides a rapid result. We must understand the common multiplication tables and the prime numbers that can totally divide the provided values.
LCM by Common Division Method
To obtain the LCM using the common division method, we must first determine the prime factors of the provided integers.
Step 1: Take the provided numbers and divide them by the least prime integer.
Step 2: In the second row, write the quotient and the number that is not divisible by the previous prime number.
Step 3: Divide the integers by the next lowest prime number.
Step 4: Keep dividing until the remaining is a prime number or one.
Step 5: To find the LCM, multiply all the divisors and the remaining prime number (if any).
Finding LCM by Long Division Method
To find an LCM by long division method let us take an example and understand this concept in a better way.
Find the LCM of 12 and 18 by the long division method.
Step one: The least prime number of the numbers 12 and 18 is 2. So we will divide both the numbers by 2.
Step two: Write the quotient below the number. The quotients further can be divided by 2.
Step three: The new quotient obtained is written below the previous one. As 9 wasn't divisible by 2 so it is written as it is in the next line.
Step four: Now we can divide 3 and 9 by 3, as it is the least prime number that can divide both 3 and 9.
Step five: We will repeat the step till 1 is obtained. Multiply all the divisors. The result would be LCM of 12 and 18.
LCM of 12 and 18 by Long Division Method
Sample Questions
1. Find the LCM of 62 and 108.
a. 3348
b. 3267
c. 4577
d. 3356
Ans: 3348
Explanation: To find the LCM using the long division method, divide 62 and 108 by the prime number and obtain a quotient that can be further divided by the prime number. Repeat the steps till you get 1 as the quotient. Multiply all the divisors to obtain LCM of 62 and 108.
Example Of LCM
2. LCM of any prime number is
a. The number is less than it.
b. The number itself
c. 1
d. None of the above
Ans: The number itself.
Explanation: The LCM of the prime number would be the number itself as the prime numbers are not divisible by any other number than by themselves. For example, 2, 3, and 5 are not divisible by any other number than themselves.
3. We get the remainder by using the division method for finding LCM.
a. True
b. False
Ans: False
Explanation: We only get the divisor and the quotients while doing the division method for finding the LCM.
Conclusion
LCM is the least common multiple and can be calculated using the division method. In this, the divisor is the prime number that would be divisible by the set of numbers given as the dividend. We need to divide till we obtain a prime number or 1. Then we can multiply all the divisors and find the value of LCM.
FAQs on LCM by Long Division Method: Step-by-Step Guide
1. What is the long division method for finding the LCM of a set of numbers?
The long division method is a systematic process to find the Least Common Multiple (LCM) of two or more numbers. In this method, the given numbers are arranged in a row and divided simultaneously by a common prime number that divides at least one of them. The process is repeated with the quotients and any undivided numbers until all the quotients become 1. The LCM is the product of all the prime divisors used.
2. What are the exact steps to find the LCM using the long division method?
To find the LCM using the long division method as per the CBSE/NCERT curriculum, follow these steps:
Step 1: Arrange the given numbers in a horizontal line, separated by commas.
Step 2: Find the smallest prime number that can divide at least one of the given numbers.
Step 3: Divide the numbers by this prime number. Write the quotient directly below the number. If a number is not divisible, write it down as it is in the next row.
Step 4: Continue this process of division with the new row of numbers until all the numbers in the last row are 1.
Step 5: The LCM is the product of all the prime numbers you used as divisors.
3. Can you show an example of finding the LCM of 12, 18, and 24 by the long division method?
Certainly. To find the LCM of 12, 18, and 24, we follow the long division steps:
First, we divide by 2: We get 6, 9, 12.
Again, divide by 2: We get 3, 9, 6 (9 is carried down as it's not divisible).
Divide by 2 once more: We get 3, 9, 3.
Now, we divide by 3: We get 1, 3, 1.
Finally, divide by 3 again: We get 1, 1, 1.
To find the LCM, we multiply all the prime divisors: 2 × 2 × 2 × 3 × 3 = 72. Therefore, the LCM of 12, 18, and 24 is 72.
4. Why is the long division method considered more efficient than other methods for finding the LCM of larger numbers?
The long division method is highly efficient for finding the LCM of three or more numbers, or larger numbers, for two main reasons. Firstly, it combines the process of finding prime factors and multiplying them into a single, organised procedure, which reduces the chance of errors. Secondly, compared to the listing multiples method, it is much faster as you don't have to write out long lists of multiples until you find a common one. It is also more straightforward than the prime factorization method for multiple numbers because you handle all numbers simultaneously.
5. What is a common mistake students should avoid when using the long division method for LCM?
A very common mistake is not carrying down a number that is not divisible by the current prime divisor. Students sometimes leave the space blank or stop processing that number. It is crucial to bring down the undivided number to the next row and continue the process. Forgetting this step will lead to an incorrect and smaller LCM, as not all factors of that number will be included in the final product.
6. Why must we use only prime numbers as divisors when calculating LCM by long division?
Using only prime numbers as divisors ensures that we are breaking down the given numbers into their fundamental building blocks. This systematic breakdown guarantees that we identify all the unique prime factors and their highest powers across all the numbers. The product of these gives the Least Common Multiple. If we used composite numbers (like 4 or 6), we might miss some prime factors or overcount others, resulting in a number that is a common multiple but not necessarily the *least* one.
7. What is the importance of learning the LCM concept in real-world applications?
The concept of LCM is very important and has several real-world applications. For instance:
Adding and Subtracting Fractions: To add or subtract fractions with different denominators, we find the LCM of the denominators to create a common denominator.
Scheduling Events: LCM helps determine when recurring events will happen at the same time again. For example, if two bells ring at intervals of 4 and 6 minutes, the LCM (12) tells us they will ring together every 12 minutes.
Purchasing Items: It can be used to figure out the minimum number of items to purchase when they come in packs of different quantities.
8. What is the fundamental difference between LCM and HCF?
The fundamental difference lies in their definitions and values. The LCM (Least Common Multiple) is the smallest number that is a multiple of all the given numbers; it is always greater than or equal to the largest of the numbers. In contrast, the HCF (Highest Common Factor), also known as the Greatest Common Divisor (GCD), is the largest number that can divide all the given numbers without leaving a remainder; it is always less than or equal to the smallest of the numbers.
