A two digit number is such that it is the product of the two distinct perfect squares. The tens digit, unit digit and the sum of these two digits are in A.P. Further if we reverse the digits mutually, it increases by 27. The oringinal number is

Options

A. a. 72
B. b. 19
C. c. 36
D. d. none of these

Answer

To solve the problem step by step, let's break down the requirements:

1. **Identify the two-digit number**: The number is a two-digit number that is the product of two distinct perfect squares.

2. **Check the digits in Arithmetic Progression (A.P.)**: The tens digit, unit digit, and the sum of these two digits must be in A.P.

3. **Reversal condition**: When the digits are reversed, the new number should be 27 more than the original number.

### Step 1: Identify the two-digit number as a product of two distinct perfect squares.

The distinct perfect squares less than 100 are:
- 1 (1^2)
- 4 (2^2)
- 9 (3^2)
- 16 (4^2)
- 25 (5^2)
- 36 (6^2)
- 49 (7^2)
- 64 (8^2)
- 81 (9^2)

Now, we can find the products of two distinct perfect squares:
- 1 * 4 = 4
- 1 * 9 = 9
- 1 * 16 = 16
- 1 * 25 = 25
- 1 * 36 = 36
- 1 * 49 = 49
- 1 * 64 = 64
- 1 * 81 = 81
- 4 * 9 = 36
- 4 * 16 = 64
- 4 * 25 = 100 (not a two-digit number)
- 9 * 16 = 144 (not a two-digit number)
- 9 * 25 = 225 (not a two-digit number)
- 16 * 25 = 400 (not a two-digit number)

The valid two-digit products of distinct perfect squares are:
- 36 (6 * 6)
- 64 (8 * 8)

### Step 2: Check the digits in A.P.

For the number 36:
- Tens digit = 3
- Units digit = 6
- Sum = 3 + 6 = 9

Now check if 3, 6, and 9 are in A.P.:
- The difference between 3 and 6 is 3.
- The difference between 6 and 9 is also 3.
- Hence, they are in A.P.

### Step 3: Check the reversal condition.

When we reverse 36, we get 63.
Now, check if 63 is 27 more than 36:
- 63 - 36 = 27

This condition is satisfied.

### Conclusion:

The original number is **36**.