`31x5` divisible by 3 ?

Options

A. `0, 3, 6` or `9`
B. `0, 4, 8` or `7`
C. `1, 3, 5` or `7`
D. `2, 4, 6` or `8`

Answer

To determine if the number \( 31x5 \) is divisible by 3, we can use the divisibility rule for 3, which states that a number is divisible by 3 if the sum of its digits is divisible by 3.

### Step-by-Step Solution:

1. **Identify the digits of the number**: The number is \( 31x5 \). The digits are 3, 1, \( x \), and 5.

2. **Calculate the sum of the digits**:
\[ \text{Sum} = 3 + 1 + x + 5 \]
Simplifying this gives:
\[ \text{Sum} = 9 + x \]

3. **Check divisibility by 3**: For \( 31x5 \) to be divisible by 3, the sum \( 9 + x \) must be divisible by 3.

4. **Evaluate possible values for \( x \)**:
- Since \( 9 \) is already divisible by 3, we need to find values of \( x \) such that \( 9 + x \) is also divisible by 3.
- We can check values of \( x \) from 0 to 9.

5. **Testing values of \( x \)**:
- If \( x = 0 \): \( 9 + 0 = 9 \) (divisible by 3)
- If \( x = 1 \): \( 9 + 1 = 10 \) (not divisible by 3)
- If \( x = 2 \): \( 9 + 2 = 11 \) (not divisible by 3)
- If \( x = 3 \): \( 9 + 3 = 12 \) (divisible by 3)
- If \( x = 4 \): \( 9 + 4 = 13 \) (not divisible by 3)
- If \( x = 5 \): \( 9 + 5 = 14 \) (not divisible by 3)
- If \( x = 6 \): \( 9 + 6 = 15 \) (divisible by 3)
- If \( x = 7 \): \( 9 + 7 = 16 \) (not divisible by 3)
- If \( x = 8 \): \( 9 + 8 = 17 \) (not divisible by 3)
- If \( x = 9 \): \( 9 + 9 = 18 \) (divisible by 3)

6. **List the valid values of \( x \)**: The values of \( x \) that make \( 31x5 \) divisible by 3 are \( 0, 3, 6, \) and \( 9 \).

### Conclusion:
The values of \( x \) for which \( 31x5 \) is divisible by 3 are \( 0, 3, 6, \) and \( 9 \).