Given Information:
- The HCF (Highest Common Factor) of two numbers is 27.
- The LCM (Least Common Multiple) of two numbers is 162.
- One of the numbers is 81.

To Find:
The other number.

Solution:

The HCF (Highest Common Factor) of two numbers is the largest number that divides both numbers without leaving a remainder. In this case, the HCF is given as 27.

The LCM (Least Common Multiple) of two numbers is the smallest number that is a multiple of both numbers. In this case, the LCM is given as 162.

We are given one of the numbers as 81. Let's denote the other number as 'x'.

Step 1: Finding the Prime Factorization of the Numbers:

To find the other number, we need to first find the prime factorization of both 27 and 162.

The prime factorization of 27 is 3 x 3 x 3 (3 raised to the power of 3).
The prime factorization of 162 is 2 x 3 x 3 x 3 x 3 (2 x 3 raised to the power of 4).

Step 2: Identifying Common Factors:

From the prime factorizations, we can identify the common factors between 27 and 162. The common factors are the prime factors that are present in both numbers.

In this case, the common factor is 3 raised to the power of 3, which is 27.

Step 3: Writing the HCF in Terms of Common Factors:

Since the HCF is 27, we can write it in terms of the common factors as follows:
HCF = 3 x 3 x 3 (27)

Step 4: Writing the LCM in Terms of Common Factors:

Next, we can write the LCM in terms of the common factors. The LCM is already given as 162, so we don't need to modify it.

Step 5: Expressing the Numbers in Terms of HCF and LCM:

The number 81 can be expressed in terms of the common factors as follows:
81 = 3 x 3 x 3 x 3 (3 raised to the power of 4)

The other number 'x' can be expressed in terms of the common factors as follows:
x = 27 x y (where y represents the remaining common factors)

Step 6: Writing the LCM in Terms of HCF and the Other Number:

We know that the LCM is equal to the product of the HCF and the other number. So, we can write the equation as follows:
LCM = HCF x x

Substituting the values:
162 = 27 x x

Simplifying the equation:
162 = 27x

Step 7: Solving for the Other Number:

To find the value of 'x', we divide both sides of the equation by 27:
162/27 = x
6 = x

Therefore, the other number is 6.

Conclusion:

The other number is 6. By following the steps outlined above, we determined that if one of the numbers is 81 and the HCF is

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