Find HCF by using the prime factor method: 54, 81 and 99.
(a) 8
(b) 9
(c) 10
(d) 11

Hint: To find the HCF (Highest Common Factor) by prime factor method, first you need to write down the prime factors of each number, check what all prime factors are common. HCF is basically the product of all the common factors from all the numbers.

Complete step-by-step answer:
We have been given the numbers 54, 81 and 99 and we need to find the HCF which means highest common factor using the prime factor method.
Let us write all the numbers and find its HCF using the prime factor method, which means we can only use prime numbers as factors.
First, let us find the factors of the number 54,
 $ 54=2\times 3\times 3\times 3 $
Now, we have to convert the right-hand side into exponential form, we get
 $ 54=2\times {{3}^{3}} $ or $ 54=2\times 3\times {{3}^{2}} $
Here, in this $ {{3}^{3}} $ , the power 3 indicates that 3 is multiplied thrice by itself.
This method is known as the prime factorization method.
Similarly, let us also find out the prime factors for the numbers 81 and 99
For 81, we get,
 $ 81=3\times 3\times 3\times 3 $
 $ 81={{3}^{4}} $ or $ 81={{3}^{2}}\times {{3}^{2}} $
Let us do the same for the number 99,
 $ \begin{align}
  & 99=11\times 3\times 3 \\
 & =11\times {{3}^{2}}
\end{align} $
In the next step, let us find the common prime factors in all the three numbers, 54, 81 and 99.
From the numbers 54, 81 and 99 the only common prime factor is $ {{3}^{2}} $ which is equal to 9.
Hence, the HCF among the numbers 54, 81 and 99 is 9

Note: In this question, when you write the prime common factors, you get only $ {{3}^{2}} $ which is 9 and no other common factor, and hence you write that down. But if you get more than 1 common factor, the multiplication of those two common factors will be your HCF. If there are no common factors among the numbers given, your HCF is 1.