Divide the L.C.M. of 36, 54 and 90 by their HCF.

To solve this question, we first need to find the LCM (Least Common Multiple) of the given numbers (36, 54, and 90) and the HCF (Highest Common Factor) of the same numbers.

Step 1: Finding the LCM

The LCM of three numbers is the smallest number that is a multiple of all three numbers. To find the LCM, we can use the prime factorization method:

36 = 2 x 2 x 3 x 3 = \(2^2 \cdot 3^2\)
54 = 2 x 3 x 3 x 3 = \(2^1 \cdot 3^3\)
90 = 2 x 3 x 3 x 5 = \(2^1 \cdot 3^2 \cdot 5^1\)

Now, we take the highest power of each prime factor present in the given numbers and multiply them together:

LCM = \(2^2 \cdot 3^3 \cdot 5^1\) = 4 x 27 x 5 = 540

Step 2: Finding the HCF

The HCF of three numbers is the largest number that divides all three numbers. We can also find the HCF using the prime factorization method. For that, we take the lowest power of the common prime factors:

HCF = \(2^1 \cdot 3^2\) = 2 x 9 = 18

Step 3: Dividing the LCM by the HCF

Now, we simply divide the LCM by the HCF:

\(\frac{LCM}{HCF} = \frac{540}{18} = 30\)

So, the answer is Option C: 30.