Let a ,b be two single digit natural numbers and 'aa bb' is a four digit number which is perfet square of natural number, such that `a+b=2k+1.` Find k.

Answer

To solve the problem, we need to find the value of \( k \) given that \( aa bb \) is a four-digit number that is a perfect square, and \( a + b = 2k + 1 \).

### Step-by-step Solution:

1. **Understanding the Number Representation**:
The four-digit number \( aa bb \) can be expressed in terms of \( a \) and \( b \):
\[ aa bb = 1000a + 100a + 10b + b = 1100a + 11b \]

2. **Factoring Out Common Terms**:
We can factor out 11 from the expression:
\[ aa bb = 11(100a + b) \]
For \( aa bb \) to be a perfect square, \( 11(100a + b) \) must also be a perfect square.

3. **Condition for Perfect Square**:
Since 11 is a prime number, for \( 11(100a + b) \) to be a perfect square, \( 100a + b \) must be of the form \( 11k^2 \) for some integer \( k \). Thus, we can write:
\[ 100a + b = 11m^2 \quad \text{for some integer } m \]

4. **Finding Values for \( a \) and \( b \)**:
Since \( a \) and \( b \) are single-digit natural numbers (from 1 to 9), we can explore possible values of \( 100a + b \) that are multiples of 11. The possible values for \( 100a + b \) must also be less than or equal to 999 (since \( aa bb \) is a four-digit number):
- The maximum value of \( 100a + b \) when \( a = 9 \) is \( 900 + 9 = 909 \).

5. **Finding Valid \( m \)**:
We can find valid values of \( m \) such that \( 11m^2 \) is a two-digit number:
- \( m = 1 \) gives \( 11 \times 1^2 = 11 \)
- \( m = 2 \) gives \( 11 \times 2^2 = 44 \)
- \( m = 3 \) gives \( 11 \times 3^2 = 99 \)
- \( m = 4 \) gives \( 11 \times 4^2 = 176 \) (too high)

So, the valid values for \( 100a + b \) are 11, 44, and 99.

6. **Finding Corresponding \( a \) and \( b \)**:
- For \( 100a + b = 11 \): \( a = 0 \) (not valid since \( a \) must be a single-digit natural number).
- For \( 100a + b = 44 \): \( a = 0 \) (not valid).
- For \( 100a + b = 99 \): \( a = 0 \) (not valid).

Since \( a \) must be a single-digit natural number, we check for valid combinations of \( a \) and \( b \) that satisfy \( a + b = 11 \).

7. **Finding \( k \)**:
We know from the problem statement that:
\[ a + b = 2k + 1 \]
Setting \( a + b = 11 \):
\[ 11 = 2k + 1 \]
Solving for \( k \):
\[ 11 - 1 = 2k \implies 10 = 2k \implies k = 5 \]

### Final Answer:
The value of \( k \) is \( 5 \).