Number S is obtained by squaring the sum of digits of a two digit number D. If difference between Sand D is 27, then the two digit number D is

Options

A. 24
B. 54
C. 34
D. 45

Answer

To solve the problem, we need to find a two-digit number \( D \) such that the difference between \( S \) (which is the square of the sum of the digits of \( D \)) and \( D \) is 27.

Let's break it down step by step:

### Step 1: Define the two-digit number \( D \)
Let \( D \) be represented as \( 10a + b \), where \( a \) is the tens digit and \( b \) is the units digit. Here, \( a \) can range from 1 to 9 (since \( D \) is a two-digit number) and \( b \) can range from 0 to 9.

### Step 2: Calculate the sum of the digits
The sum of the digits of \( D \) is given by:
\[ \text{Sum of digits} = a + b \]

### Step 3: Calculate \( S \)
According to the problem, \( S \) is obtained by squaring the sum of the digits:
\[ S = (a + b)^2 \]

### Step 4: Set up the equation based on the given condition
We know that the difference between \( S \) and \( D \) is 27:
\[ S - D = 27 \]
Substituting \( S \) and \( D \) into the equation gives:
\[ (a + b)^2 - (10a + b) = 27 \]

### Step 5: Simplify the equation
Rearranging the equation, we have:
\[ (a + b)^2 - 10a - b = 27 \]
Expanding \( (a + b)^2 \):
\[ a^2 + 2ab + b^2 - 10a - b = 27 \]
This simplifies to:
\[ a^2 + 2ab + b^2 - 10a - b - 27 = 0 \]

### Step 6: Solve for possible values of \( a \) and \( b \)
Now we can test possible values of \( a \) (1 to 9) and \( b \) (0 to 9) to find valid combinations that satisfy the equation.

#### Testing values:
1. **For \( D = 24 \)**:
- \( a = 2, b = 4 \)
- \( S = (2 + 4)^2 = 6^2 = 36 \)
- \( S - D = 36 - 24 = 12 \) (not valid)

2. **For \( D = 54 \)**:
- \( a = 5, b = 4 \)
- \( S = (5 + 4)^2 = 9^2 = 81 \)
- \( S - D = 81 - 54 = 27 \) (valid)

3. **For \( D = 34 \)**:
- \( a = 3, b = 4 \)
- \( S = (3 + 4)^2 = 7^2 = 49 \)
- \( S - D = 49 - 34 = 15 \) (not valid)

4. **For \( D = 45 \)**:
- \( a = 4, b = 5 \)
- \( S = (4 + 5)^2 = 9^2 = 81 \)
- \( S - D = 81 - 45 = 36 \) (not valid)

### Conclusion
The only valid two-digit number \( D \) that satisfies the condition \( S - D = 27 \) is:
\[ \boxed{54} \]