Answer

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**Hint:**

Here, we will first find the factors of the given numbers separately by using the method of Prime Factorization. We will then multiply the factors with highest power and find the LCM of the given numbers.

**Complete step by step solution:**

We are given the numbers 81,18 and 36.

Now we will find the factors for the numbers to find the LCM of these numbers by using the prime factorization method.

We will find the factors of 81.

We know that 81 is the square of 9 and 9 is the square of 3. So, we can write

\[81 = {9^2}\]

\[ \Rightarrow 81 = {\left( {{3^2}} \right)^2} = {3^4}\]

Thus the factors of 81 are \[{3^4}\] .

We will find the factors of 18 by using the method of prime factorization.

We can see that 18 is an even number, so dividing it by the least prime number 2. Therefore, we get

\[18 \div 2 = 9\]

Now we will divide 9 by the next least prime number 3, we get

\[9 \div 3 = 3\]

Now as we have obtained our quotient as prime number, we will not factorize it further.

Thus the factors of 18 are \[{2^1} \times {3^2}\].

We will find the factors of 36 by using the method of prime factorization.

We can see that 36 is an even number, so dividing it by the least prime number 2. Therefore, we get

\[36 \div 2 = 18\]

We can see that 18 is an even number, so dividing it by 2 again. Therefore, we get

\[18 \div 2 = 9\]

Now we will divide 9 by the next least prime number 3, we get

\[9 \div 3 = 3\]

Now as we have obtained our quotient as prime number, we will not factorize it further.

Thus the factors of 36 are \[{2^2} \times {3^2}\].

Now, we will find the LCM from these factors.

The LCM of these factors would be the highest exponent of the prime factors.

LCM of 81, 18, 36 \[ = {3^4} \times {2^2}\]

Applying the exponent on the terms, we get

\[ \Rightarrow \] LCM of 81, 18, 36 \[ = 81 \times 4\]

Multiplying the terms, we get

\[ \Rightarrow \] LCM of 81, 18, 36 \[ = 324\]

Therefore, the LCM of 81, 18, 36 is 324.

**Thus, option (C) is the correct answer.**

**Note:**

We know that the Least Common Multiple of the numbers is the smallest number that is divisible by all the numbers. LCM is the highest exponent of the prime factors. So, we will find the factors to find the LCM of numbers. Prime factorization is a method of writing the factors in terms of the power of prime numbers. Common Factor is the factor that is common to all the numbers whereas the prime factors are the factors, which is the product of the powers of the prime numbers.

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