Question

Find the HCF and the LCM of 81 and 237.

Answer 1

Hint: To find the HCF of LCM we use the common factorization method. In the case of HCF, we find the factor which is common in both the numbers and the highest amongst all the factors that divides the number. The term HCF means highest common factor meaning the factor which is common and highest amongst all of them.
The second is LCM, where we find the common factor of both the numbers by dividing the numbers till they can’t be divided anymore

Complete step-by-step answer:
Given numbers are 81 and 237.
To determine the HCF and LCM of any number, you need to factorise the number into a product of prime factors.
The number 81 is a perfect square as it is the square of prime number, 9. So, it can be expressed as a product of prime factors as,
\[81 = 3\times 3\times 3\times 3\]
The number 237 can be written as a product of prime factors as,
\[237 = 3\times 79\]
Now, HCF of any two numbers is the highest common factor to the given numbers. The common factor of 81 and 237 is 3.
So, the HCF is 3.
Now, LCM of any two numbers is the lowest common multiple that is divisible to the given numbers. So, it is the product of multiplying the highest power of each prime number.
So, the LCM is \[3^{4} \times 79 = 6399\].

Hence, the values of the HCF and LCM are given as \[3\] and \[6399\].

Note: Another method to find the both the HCF and LCM is first to find the HCF as the common division method just like shown above but after finding the HCF, to find the LCM we use the formula of product of LCM and HCF which is equal to the two numbers \[\left( A,B \right)\] given:
\[\Rightarrow LCM\times HCF=A\times B\]
Instead of using the common division method which takes more time. Placing the values in the formula, we get:
\[\Rightarrow LCM\times 3=A\times B\]
\[\Rightarrow LCM\times 3=81\times 237\]
\[\Rightarrow LCM=\dfrac{81\times 237}{3}\]
\[\Rightarrow LCM=6399\]