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The division method is a straightforward way to find the LCM by repeatedly dividing the numbers by their common prime factors until they reduce to 1. The product of the divisors gives the LCM. This method is efficient, especially for multiple numbers.
👉 Quick Links: Definition | Steps | Examples | FAQs
The division method is a technique used to find the Least Common Multiple (LCM) of a set of numbers. This method involves dividing the numbers by prime numbers that evenly divide at least one of the numbers in the set. You keep dividing the resulting quotients by appropriate prime numbers step-by-step until all the numbers are reduced to 1. The LCM is then found by multiplying all the prime divisors used during the division process. This method is efficient and especially useful when working with multiple numbers or when prime factorization is complex, making it a practical approach for finding the LCM.
The step-by-step process to find the LCM by the division method:
Write all the numbers side by side, separated by commas.
Find the smallest prime number that divides at least one of the numbers.
Divide the numbers by this prime number wherever possible.
Write the quotients below the original numbers.
For numbers that are not divisible by the prime, bring them down unchanged to the next row.
Repeat the process: find the next smallest prime number that divides at least one number in the new row.
Divide, write quotients below, and bring down non-divisible numbers as before.
Continue this step-by-step division until all the numbers in the row become 1.
Multiply all the prime divisors used in each division step.
The resulting product is the Least Common Multiple (LCM) of the original numbers.
Let’s solve some examples step by step
Example 1: Step-by-step explanation for finding the LCM of 12, 15, and 20 using the division method:
Step 1: Divide all numbers by 2 (the smallest prime dividing at least one number).
12 ÷ 2 = 6, 15 stays 15, 20 ÷ 2 = 10
Step 2: Divide the new numbers by 2 again.
6 ÷ 2 = 3, 15 stays 15, 10 ÷ 2 = 5
Step 3: Divide by 3 (next smallest prime).
3 ÷ 3 = 1, 15 stays 15, 5 stays 5
Step 4: Divide by 5.
1 stays 1, 15 ÷ 5 = 3, 5 ÷ 5 = 1
Step 5: Divide by 3.
1 stays 1, 3 ÷ 3 = 1, 1 stays 1
Step 6: Multiply all prime divisors used: 2 × 2 × 3 × 5 × 3 = 180
Result: The LCM of 12, 15, and 20 is 180
Example 2:
Step-by-step explanation for finding the LCM of 10, 18, and 20 using the division method:
Step 1: Divide all numbers by 2 (the smallest prime dividing at least one number).
10 ÷ 2 = 5, 18 ÷ 2 = 9, 20 ÷ 2 = 10
Step 2: Divide the new numbers by 2 again.
5 stays 5, 9 stays 9, 10 ÷ 2 = 5
Step 3: Divide by 3 (next smallest prime).
5 stays 5, 9 ÷ 3 = 3, 5 stays 5
Step 4: Divide by 5.
5 ÷ 5 = 1, 3 stays 3, 5 ÷ 5 = 1
Step 5: Divide by 3.
1 stays 1, 3 ÷ 3 = 1, 1 stays 1
Step 6: Multiply all prime divisors used: 2 × 2 × 3 × 5 × 3 = 180
Result: The LCM of 10, 18, and 20 is 180.
The properties of LCM by division method in point-wise format:
The LCM is always at least as large as the largest number in the set.
The division method efficiently finds the LCM without the need to list all multiples.
It uses prime numbers step-by-step to simplify calculations.
This method is especially useful for numbers with complex prime factorization.
It avoids full prime factorization of each number, saving time and reducing errors.
The stepwise division process makes it easier to handle multiple numbers.
Overall, it provides a faster and more reliable way to calculate the LCM.
FAQs on the LCM by Division Method, covering common questions:
Q1: What is the LCM by division method? The LCM by division method finds the Least Common Multiple by dividing the given numbers by prime numbers step-by-step. You continue dividing until all numbers become 1. The product of all prime divisors used is the LCM.
Q2: When is the division method preferred? This method is preferred when dealing with multiple numbers or when prime factorization is complex. It avoids listing multiples and offers a structured way to find the LCM efficiently.
Q3: What if none of the numbers are divisible by the same prime? If no prime number divides any of the numbers, it means the numbers may themselves be prime or co-prime. In such cases, the LCM is simply the product of those numbers.
Q4: Is the division method useful for large numbers? Yes, the division method helps simplify calculations for large numbers by breaking down the process stepwise rather than finding prime factors individually for each number.
Q5: Can the division method be used for any number of numbers? Yes, this method can be used for two or more numbers. It is flexible and efficient, making it practical for finding the LCM of any quantity of numbers without listing multiples.
Prime Factorization Method for LCM
Listing Multiples Method
HCF and LCM Formula
LCM of 2 and 3 Numbers
LCM and GCD
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