If x is a natural number which is a perfect square , then the number `x + sqrt(x)` must end in :

Options

A. 0 or 5
B. 0 or 1 or 9
C. 0 or 2 or 6
D. 0 or 4 or 8

Answer

To solve the problem, we need to determine the unit digit of the expression \( x + \sqrt{x} \) where \( x \) is a perfect square natural number. Let's go through the steps systematically.

### Step 1: Identify perfect square numbers
The first few perfect square natural numbers are:
- \( 1^2 = 1 \)
- \( 2^2 = 4 \)
- \( 3^2 = 9 \)
- \( 4^2 = 16 \)
- \( 5^2 = 25 \)

### Step 2: Calculate \( x + \sqrt{x} \) for each perfect square
1. **For \( x = 1 \)**:
\[ x + \sqrt{x} = 1 + \sqrt{1} = 1 + 1 = 2 \]
**Unit digit**: 2

2. **For \( x = 4 \)**:
\[ x + \sqrt{x} = 4 + \sqrt{4} = 4 + 2 = 6 \]
**Unit digit**: 6

3. **For \( x = 9 \)**:
\[ x + \sqrt{x} = 9 + \sqrt{9} = 9 + 3 = 12 \]
**Unit digit**: 2

4. **For \( x = 16 \)**:
\[ x + \sqrt{x} = 16 + \sqrt{16} = 16 + 4 = 20 \]
**Unit digit**: 0

5. **For \( x = 25 \)**:
\[ x + \sqrt{x} = 25 + \sqrt{25} = 25 + 5 = 30 \]
**Unit digit**: 0

### Step 3: Compile the results
From our calculations, we have the following unit digits:
- For \( x = 1 \): 2
- For \( x = 4 \): 6
- For \( x = 9 \): 2
- For \( x = 16 \): 0
- For \( x = 25 \): 0

### Step 4: Identify the repeating unit digits
The unit digits we found are 2, 6, and 0. Therefore, the expression \( x + \sqrt{x} \) can end in:
- 2
- 6
- 0

### Conclusion
The number \( x + \sqrt{x} \) must end in **0, 2, or 6**.

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