Find the value of Z for which the number 417Z8 is divisible by 9.

Options

A. 7
B. 6
C. 3
D. 9

Explanation

To find the value of \( Z \) for which the number \( 417Z8 \) is divisible by 9, we can follow these steps:

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

### Step 2: Write down the digits of the number
The digits of the number \( 417Z8 \) are:
- 4
- 1
- 7
- \( Z \)
- 8

### Step 3: Calculate the sum of the known digits
First, we calculate the sum of the known digits:
\[ 4 + 1 + 7 + 8 = 20 \]

### Step 4: Include \( Z \) in the sum
Now, we include \( Z \) in the sum:
\[ 20 + Z \]

### Step 5: Set up the condition for divisibility by 9
We need \( 20 + Z \) to be divisible by 9. This means:
\[ 20 + Z \equiv 0 \ (\text{mod} \ 9) \]

### Step 6: Calculate \( 20 \mod 9 \)
Now, we find \( 20 \mod 9 \):
\[ 20 \div 9 = 2 \quad \text{(remainder 2)} \]
So, \( 20 \equiv 2 \ (\text{mod} \ 9) \).

### Step 7: Set up the equation
We need:
\[ 2 + Z \equiv 0 \ (\text{mod} \ 9) \]
This simplifies to:
\[ Z \equiv -2 \ (\text{mod} \ 9) \]
or equivalently:
\[ Z \equiv 7 \ (\text{mod} \ 9) \]

### Step 8: Determine the possible values for \( Z \)
Since \( Z \) is a single digit (0 to 9), the only value that satisfies \( Z \equiv 7 \ (\text{mod} \ 9) \) is:
\[ Z = 7 \]

### Conclusion
Thus, the value of \( Z \) for which the number \( 417Z8 \) is divisible by 9 is:
\[ \boxed{7} \]