Stepwise Guide to Finding LCM for Any Set of Numbers
We first need to understand what is LCM (Least common multiple)? Why is it used? It is the lowest number that is completely divisible by each of the given numbers. LCM stands for "Least Common Multiple" which is the smallest positive integer that is a multiple of two or more integers added together. We can find the L.C.M of two or more numbers using three methods. In this article, we will dive into the basics of the LCM method and understand how to find LCM and represent it.
Full Form of LCM
How to Find LCM by Listing Method
We can find out the common multiples of two or more numbers. Out of these repeated multiples, the LCM of two numbers can be calculated.
Steps to calculate LCM by listing method:
List the initial range of each number's multiples.
Search for multiples that appear on all number lists. Write out additional multiples for each number if there aren't any common multiples in the lists.
Find the least quantity that appears on both lists.
This is the LCM number.
Let’s take the LCM method example for the above method:
Example: Find the LCM of numbers 4 and 5 by listing methods.
Ans: LCM of 4 and 5 by the listing method:
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ..
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, …
So Common Multiples are: 20, 40, …
according to the least method, the LCM of 4 and 5 is 20.
How to Find LCM by Prime Factorisation Method
Using the prime factorization method, we can find out the prime factors of the numbers.
Steps to be followed to calculate the LCM by the prime factors method is:
Step 1: Finding each number's prime factorization is the first step in computing the LCM using the prime factors approach.
Step 2: When you write each number as a product of primes.
Step 3: Now take the highest power of each prime number.
Step 4: To obtain the LCM, multiply the prime numbers.
Let’s take the LCM method example for the above method:
Example: Calculate the LCM of 50 and 100.
Ans: Steps to be followed to calculate LCM are:
Finding each number's prime factorization:
$50: 2 \times 5 \times 5$
$100: 2 \times 2 \times 5 \times 5$
When you write each number as a product of primes.
$50: 2 \times 5 \times 5$
$100: 2 \times 2 \times 5 \times 5$
Now take the highest power of each prime number. Here the highest power of 2 and 5 is 2. Therefore,
$2 \times 2 \times 5 \times 5$
To obtain the LCM, multiply the numbers
Multiplying $2 \times 2 \times 5 \times 5=100$
How to Find LCM by Division Method
LCM by Standard Division Method
To calculate the LCM of two numbers using the division method, we have to follow the steps given below:
Step 1: To find the LCM by division method, we write the given numbers in a row separately by commas, then divide the numbers by a common prime number. Find a prime number of a factor of at least one of the given numbers. Put this prime number to the left of the given numbers.
Step 2: We stop dividing after reaching the prime numbers. The product of common and uncommon prime factors is the LCM of given numbers. (It means if the prime number in step 1 is a factor of the number, divide the number by the prime and write the quotient below. If the prime number in step 1 is not a factor of the number, write the number in the row below as it is. Continue the steps until only one is left in the last row.)
Let’s take the LCM method example for the above method:
Examples: Find the least common multiple (LCM) of 6 and 12 using the division method.
Ans: Step-by-step explanation to solve LCM by the common division method is given below:
Step-by-Step Explanation to Solve LCM by Common Division Method
Solved Examples
Q1. Find the least common multiple (LCM) of 36 and 60 using the division method.
Ans: Step-by-step explanation to solve LCM by the common division method is given below:
LCM by Division Method
LCM of 36 and 60 is 2 ×2 × 3 × 3 × 5 = 180
Q2. What is the least common multiple of 980 and 9000 using the prime factorization method?
Ans: Steps to find LCM
Find the prime factorization of 980
980 = 2 × 2 × 5 × 7 × 7
Find the prime factorization of 9000
9000 = 2 × 2 × 2 × 3 × 3 × 5 × 5 × 5
Multiply each factor the greater number of times it occurs in steps i) or ii) above to find the LCM:
LCM = 2 × 2 × 2 × 3 × 3 × 5 × 5 × 5 × 7 × 7
LCM = 441000
Practice Questions:
Q1. Find the LCM of 4, 8, and 16 by using the listing method.
Ans: 16
Q2. Find the LCM of 14 and 16.
Ans: 112
Q3. Find the LCM using the prime factors method: 60 and 72.
Ans: 360
Q4. Find the least common multiple (LCM) using the division method of: 9, 12 and 36
Ans: 36
Summary
Learning about LCM can be helpful when trying to solve problems or puzzles. For example, if we have three numbers and want to find the smallest number that is a multiple of both of those numbers, we would use the LCM method. In this article, we learned about the Least Common Multiple (LCM) and how it can be used to simplify multiple problems. We also explored three methods for finding LCM, and explained how to solve it. Now that you have a better understanding of what LCM is and why it's important, continue learning more about this intriguing number in future articles!
FAQs on How to Take LCM: Methods, Steps, and Easy Examples
1. What is the Least Common Multiple (LCM) and what is its importance in mathematics?
The Least Common Multiple (LCM) of two or more numbers is the smallest positive integer that is a multiple of all the given numbers. Its importance lies in its application for solving various mathematical problems. For example, it is essential for adding or subtracting fractions with different denominators, as you need to find a common denominator, which is the LCM of the original denominators. It is also used to solve problems related to time, speed, and distance where events repeat at different intervals.
2. What are the three main methods to find the LCM of a set of numbers?
There are three primary methods to calculate the LCM. Each method has its own use case depending on the complexity of the numbers involved:
- Listing Method: This involves listing the multiples of each number until you find the first common multiple. It is simple but best for small numbers.
- Prime Factorization Method: This involves breaking down each number into its prime factors and then multiplying the highest power of all the prime factors together.
- Division Method: This is the most common and efficient method, where you divide the given numbers simultaneously by their common prime factors until the remainder for all numbers is 1. The LCM is the product of all the prime divisors.
3. Can you explain with an example how to find the LCM using the prime factorization method?
Certainly. Let's find the LCM of 12 and 18 as an example.
- First, find the prime factors of each number:
For 12, the prime factors are 2 × 2 × 3, which can be written as 2² × 3¹.
For 18, the prime factors are 2 × 3 × 3, which can be written as 2¹ × 3². - Next, identify the highest power of each unique prime factor present in the factorizations. The unique primes are 2 and 3. The highest power of 2 is 2², and the highest power of 3 is 3².
- Finally, multiply these highest powers together to get the LCM:
LCM = 2² × 3² = 4 × 9 = 36.
Thus, the LCM of 12 and 18 is 36.
4. How is the concept of LCM used in real-life situations? Can you provide an example?
LCM is frequently used to solve problems involving events that recur at regular intervals. A classic real-world example is scheduling or timing. For instance, consider two traffic lights that turn red at intervals of 60 seconds and 75 seconds, respectively. If they both turn red at the same time, the LCM of 60 and 75 will tell us after how many seconds they will simultaneously turn red again. The LCM of 60 and 75 is 300, so they will both turn red together every 300 seconds (or 5 minutes).
5. What is the step-by-step process for calculating the LCM using the division method?
The division method is a fast and reliable way to find the LCM, especially for more than two numbers. Here are the steps:
- Write down all the numbers in a row, separated by commas.
- Find the smallest prime number that can divide at least one of the given numbers.
- Divide the numbers by that prime factor. Write the quotients directly below the numbers. If a number is not divisible, write it down as it is.
- Continue this process with the smallest prime factors until all the numbers in the last row become 1.
- The LCM is the product of all the prime numbers you used for division.
For example, to find the LCM of 8, 12, and 20, you would divide by 2, then 2 again, then 3, and finally 5. The LCM would be 2 × 2 × 2 × 3 × 5 = 120.
6. What is the relationship between the HCF (Highest Common Factor) and LCM of two numbers?
There is a fundamental relationship between the HCF and LCM of any two positive integers. The product of the two numbers is always equal to the product of their HCF and LCM. The formula is:
Number 1 × Number 2 = HCF(Number 1, Number 2) × LCM(Number 1, Number 2)
This is a very important property because if you know any three of these values, you can easily calculate the fourth. For example, if you know the HCF of two numbers, you can find their LCM without using other methods, simply by rearranging the formula.
7. How do you find the LCM of fractions, and how does it differ from finding the LCM of whole numbers?
Finding the LCM of fractions involves a specific formula that uses both LCM and HCF. The method is different from that for whole numbers. To find the LCM of two or more fractions, you use the following rule:
LCM of fractions = (LCM of the Numerators) / (HCF of the Denominators)
First, you calculate the LCM of all the numerators of the fractions. Then, you calculate the HCF of all the denominators. Finally, you divide the first result by the second to get the LCM of the fractions. This ensures you find the smallest number that is a multiple of each fraction.
8. Can the LCM of two numbers ever be one of the numbers itself? Under what condition does this happen?
Yes, it is possible for the LCM of two numbers to be one of the numbers. This specific situation occurs when one number is a multiple of the other. For example, if you want to find the LCM of 6 and 12, the multiples of 6 are 6, 12, 18, ... and the multiples of 12 are 12, 24, 36, ... The first common multiple is 12. Therefore, the LCM of 6 and 12 is 12. The general rule is: if 'a' is a multiple of 'b', then the LCM(a, b) is 'a'.
9. Why does the prime factorization method work for finding the LCM? What is the logic behind choosing the highest power of each prime factor?
The logic behind the prime factorization method is based on the definition of a multiple. For a number to be a multiple of another, it must contain all of its prime factors. The LCM must be a multiple of all the numbers in a given set. Therefore, its own prime factorization must contain all the prime factors of every number in that set.
By taking the highest power of each prime factor present in any of the numbers, we ensure two things:
- It's a Common Multiple: The resulting number will have enough prime factors to be perfectly divisible by every number in the original set.
- It's the Least Multiple: By taking only the highest required power and not any more, we ensure there are no unnecessary factors, making it the smallest possible common multiple.
