If x4y5z is exactly divisible by 9, then the least value of (x+y+z) is

Options

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

Answer

To solve the problem, we need to determine the least value of \( x + y + z \) such that the number \( x4y5z \) is divisible by 9. According to the divisibility rule for 9, 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 given is \( x4y5z \). The digits are \( x, 4, y, 5, z \).

2. **Calculate the sum of the digits**: The sum of the digits can be expressed as:
\[ S = x + 4 + y + 5 + z \]
Simplifying this, we have:
\[ S = x + y + z + 9 \]

3. **Set up the divisibility condition**: For \( x4y5z \) to be divisible by 9, the sum \( S \) must be a multiple of 9. Therefore, we can write:
\[ S = x + y + z + 9 \equiv 0 \mod 9 \]

4. **Simplify the equation**: Since 9 is already a multiple of 9, we can simplify this to:
\[ x + y + z \equiv 0 \mod 9 \]
This means that \( x + y + z \) must also be a multiple of 9.

5. **Find the least value of \( x + y + z \)**: The smallest non-negative multiple of 9 is 0, but since \( x, y, z \) must be digits (0 through 9), the smallest positive multiple of 9 is 9. Thus, the least value of \( x + y + z \) that satisfies the condition is:
\[ x + y + z = 9 \]

### Conclusion:
The least value of \( x + y + z \) such that \( x4y5z \) is divisible by 9 is **9**.