Definition
LCM is called Least Common Multiple and is known by another name as Lowest Common Multiple also. By definition, LCM is a number which is the lowest common multiple that exists among all the multiples of given numbers.
General terms used in finding LCM
Multiple of a number and the common multiples of more than one number are the terms that are used often when calculating the LCM of numbers, which are discussed with examples as below.
How to find multiples of a number?
A multiple is a number obtained when a number is multiplied by any other positive integer.
What are the multiples of 7?
To get the multiples of 7, multiply 7 with any other positive integer.
Let's multiply 7 by 2
i.e. 7 × 2 = 14
∴ 14 is a multiple of 7
More multiples of 7 can be created by multiplying 7 with more positive integers such as 3, 4, 5, 6, 7, 8..... and so on
7 × 3 = 21
7 × 4 = 28
7 × 5 = 35
7 × 6 = 42
7 × 7 = 49
7 × 8 = 56
So, here, 21, 28, 35, 42, 49, 56 are also the multiples of 7.
There exist unlimited multiples of a number.
Find common multiples of numbers
Common multiples are calculated for two or more than two numbers. They are the multiples which are common among the multiples of the given numbers.
Write the common multiples of 2 and 4.
Step 1: Find out the multiples of 2 and 4 separately.
First, find multiples of 2
2 × 1 = 2
2 × 2 = 4
2 × 3 = 6
2 × 4 = 8
2 × 5 = 10
2 × 6 = 12
The multiples of 2 are 2, 4, 6, 8, 10, 12 and so on.
Next, find multiples of 4
4 × 1 = 4
4 × 2 = 8
4 × 3 = 12
4 × 4 = 16
4 × 5 = 20
4 × 6 = 24
The multiples of 4 are 4, 8, 12, 16, 20, 24 and so on.
Step 2: Find out the common multiples of 2 and 4.
4, 8 and 12 are the common multiples of 2 and 4, because the multiples 4, 8 and 12 exist in the multiples of both 2 and 4.
Methods to find LCM
LCM (Least Common Multiple or Lowest Common Multiple) is the smallest number among the common multiples of given numbers.
There are three methods to find the least common multiple.
- Listing multiple method
- Prime factorization method
- Common division method
Listing multiple method
Listing multiple method calculates LCM from the smallest common multiple from the list multiples of given numbers.
Steps to find LCM using listing multiple method
Step 1: Find out the multiples of each of the given numbers.
Step 2: Spot the common multiples from the list of multiples of each number obtained in step 1.
Step 3: Find out the smallest number among those common multiples obtained in step 2.
Find the LCM of 2 and 4 using the listing multiple method.
Step 1: Find out the multiples of 2 and 4.
Multiples of 2
2 × 1 = 2
2 × 2 = 4
2 × 3 = 6
2 × 4 = 8
2 × 5 = 10
2 × 6 = 12
The multiples of 2 are 2, 4, 6, 8, 10, 12 and so on.
Multiples of 4
4 × 1 = 4
4 × 2 = 8
4 × 3 = 12
4 × 4 = 16
4 × 5 = 20
4 × 6 = 24
The multiples of 4 are 4, 8, 12, 16, 20, 24 and so on.
Step 2: Find out the common multiples of 2 and 4.
Therefore, the common multiples are 4, 8 and 12.
Step 3: Find out the smallest number among those common multiples, that would be the LCM of 2 and 4.
Therefore, the smallest number among common multiples 4, 8 and 12 is 4.
Hence, LCM of 2 and 4 is 4.
Prime factorization method
Prime factorization method calculates LCM by taking product of the maximum number of times the prime factors of given numbers occur.
Steps to find LCM using prime factorization method
Step 1: Split each given number into its prime factors.
Step 2: Choose the common prime factors of each given number in step 1 only with the highest power and also the other non common prime factors of each number.
Step 3: Multiply all prime factors obtained in step 2.
Find the LCM of 12 and 20 using prime factorization method.
Step 1: Split 12 and 20 into its prime factors
Prime factors of 12
12 = 2 × 2 × 3 = 2² × 3¹
Prime factors of 20
20 = 2 × 2 × 5 = 2² × 5¹
Step 2: Write the common prime factor 2 with their highest power and the rest non common prime factors.
i.e. 2², 3¹ and 5¹
Step 3: Multiply all prime factors obtained in step 2.
2² × 3¹ × 5¹ = 60
Hence, LCM of 12 and 20 is 60.
Find the LCM of 18 and 24 using prime factorization method.
Step 1: Split 18 and 24 into its prime factors
Prime factors of 18
18 = 2 × 3 × 3 = 2¹ × 3²
Prime factors of 24
24 = 2 × 2 × 2 × 3 = 2³ × 3¹
Step 2: Write the common prime factors 2 and 3 with their highest power. There are no non common prime factors.
i.e. 2³ and 3²
Step 3: Multiply all prime factors.
2³ × 3² = 72
Hence, LCM of 18 and 24 is 72.
Division method
Division method calculates LCM by taking the product of all the divisors which divide the given numbers exactly.
Steps to find LCM using division method
Step 1: Write the given numbers in a row and separate them using hyphen (-).
Step 2: Divide the numbers by a prime number which can divide at least two of the given numbers exactly. Write the numbers as it is which are not divisible. Write down their quotients just below the numbers.
Repeat step 2 until the quotients can not be divided further by any prime number.
Step 3: Multiply all prime divisors and the numbers which are left in the last row.
Find the LCM of 12 and 20 using the common division method.
Step 1: Write 12 and 20 in a row separated by hyphen (-).
Step 2: Divide 12 and 20 by prime number 2 which can divide at least two of the given numbers exactly. Write quotients 6 and 10 below 12 and 20 respectively.
Repeat the step 2.
Divide 6 and 10 by prime number 2. The quotients obtained 3 and 5 can not be divided further by any prime number.
Step 3: Multiply all divisors i.e. 2, 2 and the numbers left in the last row i.e. 3, 5.
= 2 × 2 × 3 × 5 = 60
∴ LCM of 12 and 20 is 60.
Find the LCM of 18 and 24 using the division method.
Step 1: Write 18 and 24 in a row.
Step 2: Divide 18 and 24 by prime number 2 which can divide at least two of the given numbers exactly. Write quotients 9 and 12 below 18 and 24 respectively.
Repeat the step 2.
Divide 9 and 12 by prime number 3. The quotients obtained 3 and 4 can not be divided further by any prime number.
Step 3: Multiply all divisors i.e. 2, 3 and the numbers left in the last row i.e. 3, 4.
= 2 × 3 × 3 × 4 = 72
∴ LCM of 18 and 24 is 72.
Properties of LCM
Following are the main properties of LCM:
Property of co-primes
The LCM of two or more co-prime numbers is equal to their product.
LCM of 2 and 3 is 2 × 3 = 6, because 2 and 3 are co-prime numbers.
LCM of 10 and 11 is 10 × 11 = 110, because 10 and 11 are co-prime numbers.
Multiple of Number and LCM
If one number is multiple of another number then LCM is the greatest number among the two given numbers.
LCM of 18 and 9 is 18, because 18 is multiple of 9.
LCM of 18 and 54 is 54, because 54 is multiple of 18.
Property of primes
LCM of two or more prime numbers is equal to their product.
LCM of 5 and 11 is 5 × 11 = 55, because 5 and 11 are prime numbers.
LCM of 3, 11 and 13 is 3 × 11 × 13 = 429, because 3, 11 and 13 are prime numbers.
LCM always greater
The LCM is not less than any of the given numbers.
LCM of 6 and 8 is 24, which is greater than the given numbers 6 and 8.
LCM of 3 and 10 is 30, which is greater than the given numbers 3 and 10.
Relation of LCM and HCF
The LCM and HCF are related to each other using a formula which can be used to find any of the one value i.e. LCM or HCF of given numbers, if the value of one is known.
Product of LCM and HCF of a given numbers is equal to the product of the numbers.
LCM × HCF of two numbers = Product of the two numbers
