Is a two-digit integer 'x' dividible by 12?
I. When 'x' is divided by 5, the remainder is 2.
II. When 'x' is divided by 3, the remainder is 1.

Options

A. if the data in statement I alone are sufficient to answer the questin, while the data in statement II alone are not sufficient to answer the question.
B. if the data in statement II alon are sufficient to answer the question, while the data in statement I alone are not sufficient to answer the question.
C. if the data either in statement I alone or in statement II alone in sufficient to answer the question.
D. if the data even in both the statements I and II together are not sufficient to answer the question.

Explanation

To determine whether a two-digit integer \( x \) is divisible by 12, we need to analyze the two statements provided.

### Step 1: Understand the divisibility rule for 12
A number is divisible by 12 if it is divisible by both 3 and 4. Therefore, we need to check if \( x \) meets these criteria.

### Step 2: Analyze Statement I
**Statement I:** When \( x \) is divided by 5, the remainder is 2.

This means:
\[ x = 5k + 2 \]
for some integer \( k \).

However, this statement does not provide any information about the divisibility of \( x \) by 3 or 4. Therefore, we cannot conclude whether \( x \) is divisible by 12 based solely on this statement.

**Conclusion from Statement I:** Not sufficient.

### Step 3: Analyze Statement II
**Statement II:** When \( x \) is divided by 3, the remainder is 1.

This means:
\[ x = 3m + 1 \]
for some integer \( m \).

Since \( x \) leaves a remainder of 1 when divided by 3, it cannot be divisible by 3. Consequently, if \( x \) is not divisible by 3, it cannot be divisible by 12 (since 12 requires divisibility by both 3 and 4).

**Conclusion from Statement II:** Sufficient to conclude that \( x \) is not divisible by 12.

### Final Conclusion
- Statement I alone is not sufficient.
- Statement II alone is sufficient.

Thus, the answer is that **only Statement II is sufficient to determine whether \( x \) is divisible by 12**.

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