If `7xx8` is exactly divisible by `9`, then the least value of `x` is

Options

A. `0`
B. `2`
C. `3`
D. `5`

Answer

To solve the problem, we need to find the least value of \( x \) such that the number \( 7xx8 \) is exactly divisible by \( 9 \).

### Step-by-Step Solution:

1. **Understand the divisibility rule for 9**: A number is divisible by 9 if the sum of its digits is divisible by 9.

2. **Identify the digits in the number**: The number is \( 7xx8 \). The digits are \( 7 \), \( x \), \( x \), and \( 8 \).

3. **Calculate the sum of the digits**:
\[ \text{Sum} = 7 + x + x + 8 = 15 + 2x \]

4. **Set up the condition for divisibility by 9**: We want \( 15 + 2x \) to be divisible by 9.

5. **Find the smallest multiple of 9 that is greater than or equal to 15**: The multiples of 9 are \( 9, 18, 27, \ldots \). The smallest multiple of 9 that is greater than 15 is \( 18 \).

6. **Set up the equation**:
\[ 15 + 2x = 18 \]

7. **Solve for \( x \)**:
\[ 2x = 18 - 15 \]
\[ 2x = 3 \]
\[ x = \frac{3}{2} = 1.5 \]
Since \( x \) must be a single digit (0-9), we round up to the next whole number, which is \( 2 \).

8. **Check if \( x = 2 \) satisfies the condition**:
\[ 15 + 2(2) = 15 + 4 = 19 \]
\( 19 \) is not divisible by \( 9 \).

9. **Try the next integer, \( x = 3 \)**:
\[ 15 + 2(3) = 15 + 6 = 21 \]
\( 21 \) is divisible by \( 9 \).

10. **Conclusion**: The least value of \( x \) that makes \( 7xx8 \) divisible by \( 9 \) is \( 3 \).

### Final Answer:
The least value of \( x \) is \( 3 \).

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