If the 4-digit number x27y is exactly divisible by 9, then the least value of (x+y) is

Options

A. 0
B. 3
C. 6
D. 9

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) in the 4-digit number \( x27y \) such that the number is divisible by 9. The rule for divisibility by 9 states that a number is divisible by 9 if the sum of its digits is divisible by 9.

### Step-by-step Solution:

1. **Identify the digits of the number**: The number is \( x27y \), where the digits are \( x, 2, 7, \) and \( y \).

2. **Calculate the sum of the digits**: The sum of the digits can be expressed as:
\[ S = x + 2 + 7 + y = x + y + 9 \]

3. **Determine the condition for divisibility by 9**: For \( x27y \) to be divisible by 9, the sum \( S \) must also be divisible by 9. Therefore, we need:
\[ x + y + 9 \equiv 0 \, (\text{mod } 9) \]

4. **Simplify the condition**: Since \( 9 \equiv 0 \, (\text{mod } 9) \), we can simplify the condition to:
\[ x + y \equiv 0 \, (\text{mod } 9) \]
This means \( x + y \) must be a multiple of 9.

5. **Find possible values for \( x + y \)**: The smallest non-negative multiple of 9 is 0, but since \( x \) and \( y \) are digits (0 to 9), the next possible values of \( x + y \) that are non-negative and less than or equal to 18 (the maximum sum of two digits) are:
- 9
- 18

6. **Determine the least value of \( x + y \)**: The least value of \( x + y \) that satisfies the condition is:
\[ x + y = 9 \]

7. **Conclusion**: Therefore, the least value of \( x + y \) is \( 9 \).

### Final Answer:
The least value of \( x + y \) is \( 9 \).