Problem 1258

1. Start with the given expression:
$$\frac{2x}{3} - \frac{x}{6}$$

2. Find a common denominator to combine the fractions. The least common multiple of 3 and 6 is 6:
$$\frac{2x}{3} = \frac{2x \cdot 2}{3 \cdot 2} = \frac{4x}{6}$$
So the expression becomes:
$$\frac{4x}{6} - \frac{x}{6}$$

3. Combine the fractions:
$$\frac{4x - x}{6} = \frac{3x}{6}$$

4. Simplify the fraction:
$$\frac{3x}{6} = \frac{x}{2}$$

5. For $\frac{x}{2}$ to be an integer, $x$ must be even. This is because dividing an even number by 2 results in an integer.

6. Therefore, $x$ must be even. There is no requirement for $x$ to be negative, a multiple of 3, 6, or 12.

Conclusion:
The correct statement about $x$ is that it is even, but not necessarily a multiple of 3, 6, or 12.

The final answer is $\boxed{\textbf{(B)}}$