The sum of a two digit number and the number obtained by reversing its digits is a square number. How many such numbers are there?

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the problem, we need to find how many two-digit numbers exist such that the sum of the number and the number obtained by reversing its digits is a square number.

### Step-by-step Solution:

1. **Define the Two-Digit Number**:
Let the two-digit number be represented as \( xy \), where \( x \) is the digit in the tens place and \( y \) is the digit in the units place. The value of this number can be expressed as:
\[ 10x + y \]

2. **Reverse the Digits**:
When the digits are reversed, the new number becomes \( yx \). The value of this reversed number is:
\[ 10y + x \]

3. **Sum of the Two Numbers**:
The sum of the original number and the reversed number is:
\[ (10x + y) + (10y + x) = 11x + 11y = 11(x + y) \]

4. **Condition for Square Number**:
We need \( 11(x + y) \) to be a square number. For this to happen, \( x + y \) must be such that \( 11(x + y) \) is a perfect square. Since 11 is a prime number, \( x + y \) must be equal to 11 for \( 11(x + y) \) to be a perfect square (specifically \( 121 \)).

5. **Finding Valid Pairs**:
Now, we will find pairs of digits \( (x, y) \) such that:
\[ x + y = 11 \]
where \( x \) and \( y \) are digits (0-9) and \( x \) cannot be 0 since it is a two-digit number.

- If \( x = 2 \), then \( y = 9 \) (29)
- If \( x = 3 \), then \( y = 8 \) (38)
- If \( x = 4 \), then \( y = 7 \) (47)
- If \( x = 5 \), then \( y = 6 \) (56)
- If \( x = 6 \), then \( y = 5 \) (65)
- If \( x = 7 \), then \( y = 4 \) (74)
- If \( x = 8 \), then \( y = 3 \) (83)
- If \( x = 9 \), then \( y = 2 \) (92)

6. **Count the Valid Pairs**:
The valid pairs are:
- (2, 9)
- (3, 8)
- (4, 7)
- (5, 6)
- (6, 5)
- (7, 4)
- (8, 3)
- (9, 2)

This gives us a total of 8 valid two-digit numbers.

### Conclusion:
Thus, the total number of two-digit numbers such that the sum of the number and the number obtained by reversing its digits is a square number is **8**.