Finding LCM using Prime factorization and multiplication tables:
Methods for calculating the LCM of given numbers.
Lets say we have to find the LCM of two numbers 8 and 12
Then, write multiples of both the numbers,
Multiple of 8 => 8, 16 , 24, 32, 40, 48,56,64, 72, 80, 88, 96, 104, 112, 120,....
Multiple of 12 => 12, 24, 36, 48, 60, 72, 84, 96, 108, 120...
We can see from the multiplication tables of 8 and 12 that there are quite a few numbers that appear in both the tables. They are mark in green color.
They are 24, 48, 72, 96, 120 and so on.
These are all multiples of both 8 and 12. And the least among them is 24.
Hence the LCM of both these numbers is 24.
Using this method, we can find LCM of any given numbers. But this is a time taking process as we have to write down the tables for all the given numbers and sometime it can take a lot of time.
In Prime factorization, we break the number into a product of prime numbers and then we find the numbers that are common to both and the numbers that are unique to each.
The LCM will have the product of numbers that are common to both, multiplied once , and the numbers that are unique to each.
Example: Lets say we have to find the LCM of the numbers 8 and 12.
8 in terms of prime factors can be written as
8 = 2 x 2 x2
And 24 can in terms of prime factors can be written as
12 = 2 x 2 x 3
Now, for LCM write once the numbers that are common to both (marked in blue) and writing all the numbers that are unique to each. None from the prime factors of 8 and 3 from the prime factors of 12.
Therefore,
LCM of 8 and 12 = 2 x 2 x 2 x 3 = 24
Step1: Write Both numbers in a grid separated by a comma ( as done in the figure below)
Step2: Starting with the smallest prime number ( i.e. 2) , check whether the given numbers are divided by 2 or not. If a number is divisible, then divide the number and write the quotient in place of that number in the next line of the grid. Else keep the same number in the next line of the grid.
Step3: Take the next higher prime number and repeat what was done in the step2 till we get all 1 in the last row of the grid.
Example: Finding the LCM of 8 and 12
Step1: Write both the numbers 8 and 12 in the grid.
Step2: 8 is an even number. It is divisible by 2. Write the quotient obtained
(here 4 ) in the next line of the grid. The same goes for 12. In place of
12, write the quotient 6 in the next line.
Step3: Again Both 4 and 6 are even numbers and therefore divisible by two.
Divide them and write the quotient(2,3) thus obtained in the next line.
Step4: Here, 2 is divided by 2. Write the quotient 1 in the next line. But 3 is not
divided by 2 therefore write 3 as it is in the next line.
Step5: Now the numbers are 1 and 3. We don't have to touch 1. And the
number 3 is divided by3. Thus, write 3 in the left column and write the
quotient of the division of 3 by 3 in the next row.
Step6: Now, there are only 1's in the last row. We stop here.
Step7: Multiply all numbers on the left hand column (2 x 2 x2 x 3 = 24 ) to get
the LCM of 8 and 12 as 24.
To know how to find HCF read,
To know the divisibility rules, read
Methods for calculating the LCM of given numbers.
1. Using multiplication tables
Lets say we have to find the LCM of two numbers 8 and 12
Then, write multiples of both the numbers,
Multiple of 8 => 8, 16 , 24, 32, 40, 48,56,64, 72, 80, 88, 96, 104, 112, 120,....
Multiple of 12 => 12, 24, 36, 48, 60, 72, 84, 96, 108, 120...
We can see from the multiplication tables of 8 and 12 that there are quite a few numbers that appear in both the tables. They are mark in green color.
They are 24, 48, 72, 96, 120 and so on.
These are all multiples of both 8 and 12. And the least among them is 24.
Hence the LCM of both these numbers is 24.
Using this method, we can find LCM of any given numbers. But this is a time taking process as we have to write down the tables for all the given numbers and sometime it can take a lot of time.
2. Using Prime factorization
In Prime factorization, we break the number into a product of prime numbers and then we find the numbers that are common to both and the numbers that are unique to each.
The LCM will have the product of numbers that are common to both, multiplied once , and the numbers that are unique to each.
Example: Lets say we have to find the LCM of the numbers 8 and 12.
8 in terms of prime factors can be written as
8 = 2 x 2 x2
And 24 can in terms of prime factors can be written as
12 = 2 x 2 x 3
Now, for LCM write once the numbers that are common to both (marked in blue) and writing all the numbers that are unique to each. None from the prime factors of 8 and 3 from the prime factors of 12.
Therefore,
LCM of 8 and 12 = 2 x 2 x 2 x 3 = 24
3. Using prime factorization of all given the numbers in one go
Step1: Write Both numbers in a grid separated by a comma ( as done in the figure below)
Step2: Starting with the smallest prime number ( i.e. 2) , check whether the given numbers are divided by 2 or not. If a number is divisible, then divide the number and write the quotient in place of that number in the next line of the grid. Else keep the same number in the next line of the grid.
Step3: Take the next higher prime number and repeat what was done in the step2 till we get all 1 in the last row of the grid.
Example: Finding the LCM of 8 and 12
Step1: Write both the numbers 8 and 12 in the grid.
Step2: 8 is an even number. It is divisible by 2. Write the quotient obtained
(here 4 ) in the next line of the grid. The same goes for 12. In place of
12, write the quotient 6 in the next line.
Step3: Again Both 4 and 6 are even numbers and therefore divisible by two.
Divide them and write the quotient(2,3) thus obtained in the next line.
Step4: Here, 2 is divided by 2. Write the quotient 1 in the next line. But 3 is not
divided by 2 therefore write 3 as it is in the next line.
Step5: Now the numbers are 1 and 3. We don't have to touch 1. And the
number 3 is divided by3. Thus, write 3 in the left column and write the
quotient of the division of 3 by 3 in the next row.
Step6: Now, there are only 1's in the last row. We stop here.
Step7: Multiply all numbers on the left hand column (2 x 2 x2 x 3 = 24 ) to get
the LCM of 8 and 12 as 24.
To know how to find HCF read,
To know the divisibility rules, read
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