Find all possible values of x for which the 4-digit number 64y3 is divisible by 9. Also, find the numbers.

Answer

To find all possible values of \( y \) for which the 4-digit number \( 64y3 \) is divisible by 9, we will 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: Identify the digits of the number
The digits of the number \( 64y3 \) are:
- 6
- 4
- \( y \)
- 3

### Step 3: Calculate the sum of the known digits
Let's add the known digits together:
\[ 6 + 4 + 3 = 13 \]

### Step 4: Include \( y \) in the sum
Now, we include \( y \) in the sum:
\[ \text{Sum} = 13 + y \]

### Step 5: Set up the condition for divisibility by 9
We need \( 13 + y \) to be divisible by 9. This means we can write:
\[ 13 + y \equiv 0 \mod 9 \]

### Step 6: Calculate \( 13 \mod 9 \)
Calculating \( 13 \mod 9 \):
\[ 13 \div 9 = 1 \quad \text{(remainder 4)} \]
So,
\[ 13 \equiv 4 \mod 9 \]

### Step 7: Set up the equation
Now we need:
\[ 4 + y \equiv 0 \mod 9 \]
This can be rearranged to:
\[ y \equiv -4 \mod 9 \]
Since \(-4\) is equivalent to \(5\) in modulo \(9\) (because \( -4 + 9 = 5 \)), we have:
\[ y \equiv 5 \mod 9 \]

### Step 8: Determine possible values for \( y \)
Since \( y \) is a single digit (0 to 9), the only possible value for \( y \) that satisfies this condition is:
\[ y = 5 \]

### Step 9: Write the final number
Substituting \( y \) back into the number \( 64y3 \):
\[ 64y3 = 6453 \]

### Conclusion
The only possible value of \( y \) for which the number \( 64y3 \) is divisible by 9 is \( 5 \), and the number is \( 6453 \).

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