- Aditi
- 26. September 2022
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The full form of LCM is the least common multiple. In mathematics, the least common multiple (LCM) is a method of finding the least common number between two numbers that is divisible by both numbers. LCM can be calculated for two or more numbers. LCM is also commonly referred to as Least Common Divisor (LCD).
There are three methods for determining the LCM of a given number: the listing method, the prime factorization method, and the division method. In this article, let's discuss all about how to find the LCM of a specific number with solved examples. Scroll down to find out more.
What is the least common multiple (LCM) in mathematics?
The smallest positive number that is a multiple of two or more numbers.
LCM-Definition
For example, LCM of 4 and 9 is 2 * 2 * 3 * 3 = 36.
Here 4 is expressed as 2*2 and 9 as 3*3.
If we consider the multiples of 4 and 9, we get:
- multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 26, …
- multiples of 9: 9, 18, 27, 36, …
The first common multiple for 4 and 9 is 36. Hence the LCM of 4 and 9 is 36.
LCM is used for adding and subtracting two fractions. When the denominator value of the fractions are not equal, LCM is used to make the denominators equal. This simplifies the whole calculation process.
Properties of LCM
There are certain properties of LCM that you should learn before knowing how to find LCM.
1. LCM Associative Property
The associative property of LCM states that LCM of A and B is the same as LCM of B and A.
LCM (A, B) = LCM (B, A)
For example, consider A as 6 and B as 2. Here 6 can be expressed as 2 * 3 and 2 as 2.
Also 6 * 2 = 2 * 3 * 2.
Now we write the numbers in exponent form and multiply the factors by the highest power.
If we write the number in exponent form, we get:
6 = 2 * 3 = 21* 31
2 = 21
So the LCM of 6 and 2 is 21* 31= 6
And LCM of 2 and 6 is also 6.
Thus it is proved that LCM(6, 2) = LCM(2, 6) = 6.
2. Commutative property of LCM
The commutative property is used when trying to find the LCM of 3 numbers. Commutative property of LCM says that
LCM (A, B, C) = LCM (LCM (A, B), C) = LCM (A, LCM (B, C))
For example, consider A as 3, B as 6, and C as 12. Here, 3 can be expressed as 3, 6 can be expressed as 2*3, and 12 can be expressed as 2*2*3.
Now we write the numbers in exponent form and multiply the factors by the highest power.
If we write the number in exponent form, we get:
3 = 31
6 = 2 * 3 = 21* 31
12 = 2 * 2 * 3 = 22* 31
So the LCM of 3, 6, and 12 is 22* 31= 2 * 2 * 3 = 12
Now the LCM of A and B, i.e. LCM 3 and 6 equals 31* 21= 6 and
LCM of (A,B) and C, i.e. H. LCM of 6 and 12 is 22* 31= 2 * 2 * 3 = 12.
LCM (LCM (A, B), C) = LCM (LCM (3, 6), 12) = LCM (6, 12) = 12
LCM of B and C, i.e. H. LCM of 6 and 12 is 22* 31= 2 * 2 * 3 = 12 and
LCM of A and LCM of (B,C), i. H. LCM of 3 and 12 is 22* 31= 2 * 2 * 3 = 12.
LCM (A, LCM (B, C)) = LCM (3, LCM (6, 12)) = LCM (3, 12) = 12.
Thus it is proved that LCM(3, 6, 12) = LCM(LCM(3, 6), 12) = LCM(3, LCM(6, 12)) = 12.
3. LCM's Distribution Property
The distributive law is also used when trying to find the LCM of 3 numbers. LCM's distribution property states that
LCM (dA, dB, dC) = d * LCM (A, B, C)
For example, consider A as 5, B as 8, C as 13, and d as any random variable.
Now we write the numbers in exponent form and multiply the factors by the highest power.
If we write the number in exponent form, we get:
5 = 51
8 = 2 * 2 * 2 = 23
13 = 131
LCM of 5, 8 and 13 is 51* 23* 131= 5 * 2 * 2 * 2 * 13 = 520.
Also LCM (5d, 8d, 13d) = d * LCM (5, 8, 13) = 520
Thus it is proved that LCM(5d, 8d, 13d) = d * LCM(5, 8, 13)
How do I find the LCM?
There are three main ways to find the LCM of two or more numbers. The methods are:
1. Split method
To find the LCM using the division method, divide the given numbers by the smallest prime divisible by each of the given numbers. Then the further obtained prime factors are used to calculate the final LCM.
You can follow the steps below to find the LCM using the split method:
- Step 1:Write all the given numbers for which you need to find the LCM, separated by commas.
- Step 2:Now find the smallest prime number that is divisible by either of the two given numbers.
- Step 3:If a number is not divisible, write that number on the next line directly below and continue.
- Step 4:Keep dividing the numbers obtained after each step by the prime numbers until you get 1 in the entire series.
- Step 5:Now multiply all the prime numbers and the end result is the LCM of the given numbers.
For example, you need to find the LCM of 12 and 5 using the division method.
| prime factors | first number | second number |
| 2 | 12 | 5 |
| 2 | 6 | 5 |
| 3 | 3 | 5 |
| 5 | 1 | 5 |
| 1 | 1 |
So LCM of 12 and 5 = 2 * 2 * 3 * 5 = 50
2. Prime factorization method
To find the LCM of given numbers using the prime factorization method, do the following:
- Step 1:Find the prime factors of the given numbers using the repeated division method explained above.
- Step 2:Write down the prime factors in their exponential forms. Then multiply the prime factors by the highest power.
- Step 3:The final result after multiplication is the LCM of the given numbers.
For example, you need to find the LCM of 18, 10, and 7 using the prime factorization method.
- Prime factorization of 18can be expressed as 2 * 3 * 3 = 21* 32
- Prime factorization of 10can be expressed as 2 * 5 = 21* 51
- The prime factorization of 7 can be expressed as 71
So the LCM of 18, 10, and 7 = 21* 32* 51* 71= 2 * 3 * 3 * 5 * 7 = 630.
3. Listing method
To find the LCM of the specified numbers using the listing method, you can follow the steps below:
- Step 1:Write down the first multiples of the given numbers separately.
- Step 2:Of all the multiples of the numbers, focus on the multiples that are common to all given numbers.
- Step 3:Now find out from all the common multiples, take out the least common multiple. That will be the LCM of the given numbers.
For example, you need to find the LCM of 8 and 5 using the listing method.
- multiples of 8are 8, 16, 24, 32, 40, 48, 64, …
- Multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 45, ...
Here it is clear that the least common multiple is 40.
So the LCM of 8 and 5 is 40.
Important LCM formulas
There are two main LCM formulas, one for finding the LCM of integers and the other for finding the LCM of fractions.
Before proceeding and knowing the formulas, you should know the HCF (Highest Common Factor).
HCF is the highest factor common among the factors of all numbers given. It is also known as the greatest common divisor (ggT).
1. Formula to find the LCM of the given integers
Let A and B be two given integers. So the LCM of A and B can be calculated with the formula:
LCM (A, B) = (A * B) / HCF (A, B),
where HCF is the highest common divisor or greatest common divisor of A and B.
Another formula to find the LCM of the given integers is:
A*B = LCM(A,B)*HCF(A,B), that is,
The product of the two integers given is equal to the product of their LCM and HCF.
2. Formula for finding the LCM of the given fractions
LCM = LCM of the numerator / HCF of the denominator
LCM-List
| LCM of 3 and 9 |
| LCM of 3 and 7 |
| LCM of 7 and 9 |
| LCM of 3 and 5 |
| LCM of 7 and 8 |
| LCM of 48 and 56 |
| LCM of 120 and 144 |
| LCM of 4 and 10 |
| LCM of 8 and 20 |
| LCM of 6 and 10 |
| LCM of 30 and 35 |
| LCM of 3 and 8 |
| LCM of 9 and 12 |
| LCM of 4 and 6 |
| LCM of 80, 85 and 90 |
| LCM of 45, 60 and 75 |
| LCM of 63, 70 and 77 |
Solved example problems based on LCM
Question 1: Find the LCM of 9 and 4 using the listing method.
Solution:
Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ...
Multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
Here it is clear that the least common multiple is 36.
So the LCM of 9 and 4 is 36.
Question 2: What is the LCM of 16 and 21 using the prime factorization method?
Solution:
The prime factorization of 16 can be expressed as 2 * 2 * 2 * 2 = 24
The prime factorization of 21 can be expressed as 3 * 7 = 31* 71
So the LCM of 16 and 21 = 24* 31* 71= 2 * 2 * 2 * 2 * 3 * 7 = 336.
Question 3: If LCM and HCF of two numbers, 5 and B, are 45 and 1 respectively. find B
Solution:
As we know,
Product of two numbers = LCM * HCF
that is given to us
One of the numbers = 5, LCM = 45 and HCF = 1
Also 5 * B = 45 * 1
B = (45 * 1) / 5
B = 9
So the other number is 9.
Question 4: Find the LCM of 24 and 45 using the division method.
Solution:
| prime factors | first number | second number |
| 2 | 24 | 45 |
| 2 | 12 | 45 |
| 2 | 6 | 45 |
| 3 | 3 | 45 |
| 3 | 1 | fifteen |
| 5 | 1 | 5 |
| 1 | 1 |
The LCM of 24 and 45 = 2 * 2 * 2 * 3 * 3 * 5 = 360.
Question 5: Find the LCM of 14, 22 and 18 using the prime factorization method.
Solution:
The prime factorization of 14 can be expressed as 2 * 7 = 21* 71
I can express the prime factorization of 22 as 2 * 11 = 21* 111
The prime factorization of 18 can be expressed as 2 * 3 * 3 = 21* 32
So the LCM of 14, 22 and 18 = 21* 71* 111* 32= 2 * 7 * 11 * 3 * 3 = 1386.
Question 6: If the HCF of two numbers, 42 and 9, is 3. Find the LCM.
Solution:
As we know,
LCM (A, B) = (A * B) / HCF (A, B)
that is given to us
Numbers = 42 and 9 and HCF = 3
Also LCM (42, 9) = (42 * 9) / HCF (42, 9)
= 378 / 3
= 126
Therefore, the LCM of 42 and 9 is 126.
Frequently asked questions about LCM
What do you mean by LCM?
LCM is the least common multiple. It is used to find the lowest possible common number divisible by all the numbers you need to find the LCM for.
How are LCM and HCF related?
LCM (A, B) = (A * B) / HCF (A, B),
where A and B are two integers.
This formula is used to find the LCM of the given integers.
Can LCM be calculated for just 2 numbers?
No, LCM can also be calculated for more than two numbers. There should be at least 2 numbers to locate the LCM.
What methods are there to find the LCM?
There are three main ways to find the LCM of two or more numbers. The methods are:
1. Split method
2. Prime factorization method
3. Listing method
What is the LCM of 2 and 8?
LCM of 2 and 8 is 8.
What are the properties of LCM?
There are three main characteristics of LCM. They are:
1. Associative ownership
2. Commutative property
3. Distribution property
What is the formula to find the LCM of the fraction?
The formula for finding the LCM of a given fraction is:
LCM = LCM of the numerator / HCF of the denominator
We hope this article about LCM is useful to you. If you have any questions about this post, ping us via the comments section below and we'll get back to you as soon as possible.
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FAQs
How do you find the LCM definition? ›
LCM denotes the least common factor or multiple of any two or more given integers. For example, L.C.M of 16 and 20 will be 2 x 2 x 2 x 2 x 5 = 80, where 80 is the smallest common multiple for numbers 16 and 20. Now, if we consider the multiples of 16 and 20, we get; 16 → 16, 32, 48, 64, 80,…
What are the 3 methods of LCM? ›- Listing the Multiples Method. To find the LCM using multiples, list the multiples of the numbers in the table as shown. ...
- Prime Factorization Method. The prime factorization method involves finding the prime factors of the given numbers and identifying the least common multiple (LCM). ...
- Division Method. ...
- Solved Examples.
Explanation: To find the least common denominator, list out the multiples of both denominators until you find the smallest multiple that is shared by both. Because 20 is the first shared multiple of 4 and 5, it must be the least common denominator for these two fractions.
What is the first step in finding the LCM *? ›The first thing we have to do is break down the prime factors of each number. After, we will have to choose the common factors and not the greatest common to the greatest exponent, and finally, we have to multiple the chosen factors. We are going to look at an example of this, calculating the LCM of 12 and 8.
How do you explain lowest common denominator? ›The least common denominator is the smallest number of all the common multiples of the denominators when 2 or more fractions are given.
What is the fastest way to find a common denominator? ›The easiest way to find a common denominator for a pair of fractions is to multiply the numerator and denominator of each fraction by the denominator of the other.
What is LCM definition and example? ›The full form of LCM in Maths is the Least common multiple. LCM is the smallest number which is divisible by two or more given numbers. For example, LCM of 2 & 3 is 6.
What is the LCM of 24 and 36? ›LCM of 24 and 36 is 72. In Maths, the LCM of any two numbers is the value which is evenly divisible by the given two numbers. Least common multiple of 24 and 36 is the smallest number we get among the common multiples. (24, 48, 72, 96, ….)
What is the LCM of 8 and 12? ›The LCM of 8 and 12 is 24.
What is a LCM in math grade 7? ›The least common multiple is the smallest whole number that is a multiple of each of two or more numbers.
What is the LCM of 4 and 6? ›
The lowest common multiple is the lowest multiple shared by two or more numbers. For example, common multiples of 4 and 6 are 12, 24 and 36, but the lowest of those is 12; therefore, the lowest common multiple of 4 and 6 is 12.