If `3x, (3)/(x), and (15)/(x)` are integers, which of the following must also be an integers?
I. `(x)/(3)`
II. X
III. `6x`

Options

A. II only
B. III only
C. I and III only
D. I, II and III

Answer

To solve the problem, we need to analyze the given conditions and determine which of the options must also be an integer if \(3x\), \(\frac{3}{x}\), and \(\frac{15}{x}\) are integers.

### Step-by-step Solution:

1. **Understanding the Conditions**:
- We know that \(3x\) is an integer. This implies that \(x\) must be a rational number such that when multiplied by 3, it results in an integer. Therefore, \(x\) can be expressed as \(\frac{k}{3}\) where \(k\) is an integer.
- Next, \(\frac{3}{x}\) is also an integer. This means that \(x\) must be a divisor of 3. The possible integer values for \(x\) can be \(1, -1, 3, -3\) (and any rational number that satisfies this condition).
- Finally, \(\frac{15}{x}\) being an integer means that \(x\) must also be a divisor of 15. The integer divisors of 15 are \(1, -1, 3, -3, 5, -5, 15, -15\).

2. **Finding Common Divisors**:
- The common integer divisors of both 3 and 15 are \(1, -1, 3, -3\). Hence, \(x\) must be one of these values.

3. **Evaluating Each Option**:
- **Option I: \(\frac{x}{3}\)**:
- If \(x = 3\), then \(\frac{x}{3} = \frac{3}{3} = 1\) (an integer).
- If \(x = -3\), then \(\frac{x}{3} = \frac{-3}{3} = -1\) (an integer).
- If \(x = 1\), then \(\frac{x}{3} = \frac{1}{3}\) (not an integer).
- If \(x = -1\), then \(\frac{x}{3} = \frac{-1}{3}\) (not an integer).
- Thus, \(\frac{x}{3}\) is not guaranteed to be an integer for all valid \(x\).

- **Option II: \(x\)**:
- Since \(x\) can take values \(1, -1, 3, -3\), all of these values are integers. Therefore, \(x\) must be an integer.

- **Option III: \(6x\)**:
- If \(x = 3\), then \(6x = 6 \times 3 = 18\) (an integer).
- If \(x = -3\), then \(6x = 6 \times -3 = -18\) (an integer).
- If \(x = 1\), then \(6x = 6 \times 1 = 6\) (an integer).
- If \(x = -1\), then \(6x = 6 \times -1 = -6\) (an integer).
- Thus, \(6x\) is guaranteed to be an integer for all valid \(x\).

### Conclusion:
From the analysis, we find that:
- Option I: Not necessarily an integer.
- Option II: Must be an integer.
- Option III: Must be an integer.

Thus, the correct answer is that options II and III must be integers.

### Final Answer:
**II and III must be integers.**