When a two digit number is subtracted from the other two digit number which consists of the same digits but in reverse order, then the difference comes out to be a two digit perfect square. The number is :

Options

A. 59
B. 73
C. 36
D. not unique

Answer

To solve the problem, we need to find a two-digit number such that when it is subtracted from its reverse, the difference is a two-digit perfect square. Let's denote the two-digit number as \(10x + y\), where \(x\) is the tens digit and \(y\) is the units digit.

### Step-by-Step Solution:

1. **Define the Two-Digit Number**:
Let the two-digit number be \(N = 10x + y\), where \(x\) and \(y\) are the digits of the number.

2. **Define the Reverse of the Number**:
The reverse of the number \(N\) is \(M = 10y + x\).

3. **Calculate the Difference**:
We need to find the difference \(D\) when \(N\) is subtracted from \(M\):
\[ D = M - N = (10y + x) - (10x + y) = 10y + x - 10x - y = 9y - 9x = 9(y - x) \]

4. **Condition for Perfect Square**:
The difference \(D\) must be a two-digit perfect square. Thus, we have:
\[ D = 9(y - x) \]
For \(D\) to be a two-digit perfect square, \(y - x\) must be such that \(9(y - x)\) is a two-digit perfect square.

5. **List Two-Digit Perfect Squares**:
The two-digit perfect squares are: \(16\) (4²), \(25\) (5²), \(36\) (6²), \(49\) (7²), \(64\) (8²), \(81\) (9²).

6. **Find Possible Values for \(y - x\)**:
We can find the values of \(y - x\) that yield two-digit perfect squares:
- \(9(y - x) = 16 \Rightarrow y - x = \frac{16}{9}\) (not an integer)
- \(9(y - x) = 25 \Rightarrow y - x = \frac{25}{9}\) (not an integer)
- \(9(y - x) = 36 \Rightarrow y - x = 4\)
- \(9(y - x) = 49 \Rightarrow y - x = \frac{49}{9}\) (not an integer)
- \(9(y - x) = 64 \Rightarrow y - x = \frac{64}{9}\) (not an integer)
- \(9(y - x) = 81 \Rightarrow y - x = 9\)

7. **Possible Cases**:
From the above calculations, we have two possible cases:
- \(y - x = 4\)
- \(y - x = 9\)

8. **Finding Valid Two-Digit Numbers**:
- For \(y - x = 4\):
- Possible pairs \((x, y)\): \((1, 5), (2, 6), (3, 7), (4, 8), (5, 9)\)
- Corresponding numbers: \(15, 26, 37, 48, 59\)
- For \(y - x = 9\):
- Possible pair \((x, y)\): \((0, 9)\) (which gives 09, not a valid two-digit number)

9. **Check the Valid Numbers**:
- Check \(37\):
- Reverse: \(73\)
- Difference: \(73 - 37 = 36\) (which is a perfect square)
- Check \(59\):
- Reverse: \(95\)
- Difference: \(95 - 59 = 36\) (which is also a perfect square)

10. **Conclusion**:
The two-digit number that satisfies the condition is **73**.