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What if you were asked to find the HCF and
LCM of 16 and 20?
You have two options.
Either you find the HCF and LCM separately.
Or we find them together.
Of course we will find them together.
All we need to do
is use the common division method of finding the LCM.
In the first step,
we write the numbers separated by a comma.
And then draw an L shaped line like this.
Then we look for a common factor of the two numbers,
and write it here.
2 is common factor of both the numbers.
Then we write the quotients underneath.
2 times 8 is 16.
And 2 times 10 is 20.
Since these numbers are not co-prime
we go back to the second step and continue the process.
They have two as the common factor.
And 2 times 4 is 8.
2 times 5 is 10.
We stop here since 4 & 5 are co-prime numbers.
They have no common factors except 1.
The product of the numbers in this L shape
gives us the LCM of the two numbers.
The LCM equals 2 times 2, times 4, times 5.
That is 4 multiplied by 20 which equals 80.
Yes,
but how does this give us the HCF?
Well,
the product of the factors on the left gives us the HCF.
Hence, the HCF will equal 2 times 2 which is 4.
This is the HCF of 16 and 20.
So you can see that just this one method
gave us the HCF as well as the LCM of two numbers.
But we also have to note that this works only for two numbers.
There is another interesting concept above the HCF
and the LCM of two numbers.
First, let's find the product of the LCM and the HCF.
LCM is 80 and the HCF is 4.
The product will be 80 times 4 which equals 320.
Now let's find the product of the two numbers.
It will be 16 times 20
which equals 320.
This will always be true!
For any two given numbers,
the product of their LCM and HCF
will be equal to the product of the two given numbers.
