Find the value of z for which the number 471z8 is divisible by 9. Also find the number.

Answer

To find the value of \( z \) for which the number \( 471z8 \) 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 \( 471z8 \) are 4, 7, 1, \( z \), and 8.

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

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

### Step 5: Set up the equation for divisibility by 9
We need \( 20 + z \) to be a multiple of 9. We can express this as:
\[ 20 + z \equiv 0 \mod 9 \]

### Step 6: Find the remainder of 20 when divided by 9
Calculating \( 20 \mod 9 \):
\[ 20 \div 9 = 2 \quad \text{(remainder 2)} \]
So, \( 20 \equiv 2 \mod 9 \).

### Step 7: Set up the equation to solve for \( z \)
We need:
\[ 2 + z \equiv 0 \mod 9 \]
This simplifies to:
\[ z \equiv -2 \mod 9 \]
which is equivalent to:
\[ z \equiv 7 \mod 9 \]

### Step 8: Determine the value of \( z \)
Since \( z \) must be a single digit (0 to 9), the only solution is:
\[ z = 7 \]

### Step 9: Write the complete number
Substituting \( z \) back into the number gives us:
\[ 47178 \]

### Step 10: Verify divisibility
Now, let's verify if \( 47178 \) is divisible by 9:
\[ 4 + 7 + 1 + 7 + 8 = 27 \]
Since \( 27 \) is divisible by 9, our solution is confirmed.

### Final Answer
The value of \( z \) is \( 7 \) and the number is \( 47178 \).

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