Correct Option (Option 4):

The question asks whether x is an integer. To determine sufficiency, we must ascertain if the given statements, individually or combined, provide a definitive 'yes' or 'no' answer to this question.

Analysis of Statement-1: $\frac{x}{3}$ is not an integer.
- If x = 7, then $\frac{x}{3} = \frac{7}{3}$ , which is not an integer. In this instance, x is an integer.
- If x = $\frac{7}{3}$ , then $\frac{x}{3} = \frac{7}{9}$ , which is not an integer. In this instance, x is not an integer.
Since Statement-1 permits scenarios where x is an integer and where x is not an integer, it is not sufficient to answer the question.
Analysis of Statement-2: 3x is an integer.
- Let 3x = k, where k represents an integer. This implies that $x = \frac{k}{3}$ .
- If k = 21 (an integer), then $x = \frac{21}{3} = 7$ . In this instance, x is an integer.
- If k = 7 (an integer), then $x = \frac{7}{3}$ . In this instance, x is not an integer.
Since Statement-2 also allows for both possibilities, it is not sufficient to answer the question.
Combined Analysis of Statement-1 and Statement-2:
- From Statement-2, we establish that $x = \frac{k}{3}$ for some integer k.
- Substituting this into Statement-1 yields $\frac{x}{3} = \frac{k / 3}{3} = \frac{k}{9}$ .
- Statement-1 asserts that $\frac{k}{9}$ is not an integer. This condition implies that k is not a multiple of 9.
- Now, we evaluate whether x (which is k3) is an integer, given that k is an integer and not a multiple of 9.
  - Consider k = 21. Here, k is an integer and not a multiple of 9. Then $x = \frac{21}{3} = 7$ , which is an integer.
  - Consider k = 7. Here, k is an integer and not a multiple of 9. Then $x = \frac{7}{3}$ , which is not an integer.
As both statements combined still lead to scenarios where x can be an integer or not an integer, they are not sufficient to provide a definitive answer to the question.

Therefore, both Statement 1 and Statement 2 are not sufficient to answer the question.

Incorrect Options:

Option 1: Statement 1 alone is sufficient to answer the question.

This is incorrect. As demonstrated, Statement 1 allows for x to be an integer (e.g., x=7) and for x not to be an integer (e.g., x= $\frac{7}{3}$ ). Thus, it does not provide a conclusive answer.
Option 2: Statement 2 alone is sufficient to answer the question.

This is incorrect. As demonstrated, Statement 2 allows for x to be an integer (e.g., x=7) and for x not to be an integer (e.g., x= $\frac{7}{3}$ ). Thus, it does not provide a conclusive answer.
Option 3: Both Statement 1 and 2 are sufficient to answer the question.

This is incorrect. Even when both statements are considered together, counterexamples exist (x=7 and x= $\frac{7}{3}$ ) that satisfy both conditions but yield different answers to the question. Therefore, the combined statements are not sufficient.