Find all possible values of x for which the 4-digit number 320x is divisible by 3. Also, find the numbers.

Answer

To solve the problem, we need to find all possible values of \( x \) such that the 4-digit number \( 320x \) is divisible by 3.

### Step-by-Step Solution:

1. **Understand the Divisibility Rule for 3**:
A number is divisible by 3 if the sum of its digits is divisible by 3.

2. **Identify the Digits of the Number**:
The digits of the number \( 320x \) are 3, 2, 0, and \( x \).

3. **Calculate the Sum of the Known Digits**:
First, we calculate the sum of the known digits:
\[ 3 + 2 + 0 = 5 \]
Therefore, the total sum of the digits becomes:
\[ 5 + x \]

4. **Set Up the Condition for Divisibility by 3**:
We need \( 5 + x \) to be divisible by 3. This means we need to find values of \( x \) such that:
\[ 5 + x \equiv 0 \ (\text{mod} \ 3) \]

5. **Calculate the Remainder of 5 when Divided by 3**:
The remainder when 5 is divided by 3 is:
\[ 5 \div 3 = 1 \quad \text{(remainder 2)} \]
Thus, \( 5 \equiv 2 \ (\text{mod} \ 3) \).

6. **Determine the Values of \( x \)**:
We need \( 5 + x \equiv 0 \ (\text{mod} \ 3) \), which simplifies to:
\[ 2 + x \equiv 0 \ (\text{mod} \ 3) \]
This means:
\[ x \equiv 1 \ (\text{mod} \ 3) \]
The possible values of \( x \) that satisfy this condition (considering \( x \) must be a single digit, i.e., 0 to 9) are:
- \( x = 1 \)
- \( x = 4 \)
- \( x = 7 \)

7. **Form the Numbers**:
Now we can form the numbers using these values of \( x \):
- For \( x = 1 \): The number is \( 3201 \)
- For \( x = 4 \): The number is \( 3204 \)
- For \( x = 7 \): The number is \( 3207 \)

### Conclusion:
The possible values of \( x \) for which the number \( 320x \) is divisible by 3 are \( 1, 4, \) and \( 7 \). The corresponding numbers are \( 3201, 3204, \) and \( 3207 \).