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The question asks whether \( x \) is an integer, given the following two statements:

- **S1**: \( \frac{x}{3} \) is not an integer.
- **S2**: \( 3x \) is an integer.

We need to determine whether the statements are sufficient to conclude that \( x \) is an integer, using the following options:
- (a) S1 alone is sufficient.
- (b) S2 alone is sufficient.
- (c) Both S1 and S2 are required.
- (d) Both S1 and S2 are not sufficient.

### Analysis:

1. **S1: \( \frac{x}{3} \) is not an integer**  
   This tells us that \( x \) is not divisible by 3. However, it doesn't provide enough information to confirm whether \( x \) is an integer. For example, \( x = 5 \) would satisfy \( \frac{5}{3} \) not being an integer, but \( x = \frac{5}{2} \) would also satisfy this, and \( \frac{5}{2} \) is not an integer.  
   **S1 alone is not sufficient.**

2. **S2: \( 3x \) is an integer**  
   This means that \( x \) is a rational number, because if \( 3x \) is an integer, then \( x \) could be any number that, when multiplied by 3, gives an integer. For example, \( x = 1 \) would satisfy this, but so would \( x = \frac{1}{3} \), which is not an integer.  
   **S2 alone is also not sufficient.**

3. **Combining S1 and S2**  
   If we know that \( \frac{x}{3} \) is not an integer (S1) and that \( 3x \) is an integer (S2), then \( x \) cannot be a non-integer multiple of 3 (like \( \frac{1}{3} \)). Therefore, the only possible solution is that \( x \) is an integer.  
   **Both S1 and S2 together are sufficient to conclude that \( x \) is an integer.**

### Correct Answer:
**(c) Both required.**

Would you like further details or have any questions?

Here are 5 related questions to extend the discussion:
1. What happens if \( \frac{x}{3} \) is an integer?
2. Can you provide an example where \( S1 \) holds but \( S2 \) doesn't?
3. How can you prove that a number is an integer from the given conditions?
4. Is it possible for both \( \frac{x}{3} \) and \( 3x \) to be non-integers?
5. Can this method be generalized to other divisibility problems?

**Tip**: In data sufficiency problems, always check if combining the statements provides additional insight that individual statements lack.

Is x an integer? S1: x/3 is not an integer. S2: 3x is an integer.

Determine whether x is an integer using the conditions: S1: x/3 is not an integer, and S2: 3x is an integer. Explore divisibility, rational numbers, and integer properties in this algebraic problem. Suitable for students in Grades 9-12.

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