Suppose x is divisible by 8 but not by 3. Then which of the following CANNOT be an integer?

Options

A. `x//2`
B. `x//4`
C. `x//6`
D. `x//8`

Answer

To solve the problem, we need to analyze the conditions given: \( x \) is divisible by 8 but not by 3. We will evaluate the divisibility of \( x \) with respect to the options provided.

### Step-by-Step Solution:

1. **Understanding Divisibility by 8**:
Since \( x \) is divisible by 8, we can express \( x \) as:
\[ x = 8k \]
for some integer \( k \).

2. **Divisibility by 2 and 4**:
Because \( x \) is divisible by 8, it is also divisible by 2 and 4. Therefore:
\[ \frac{x}{2} = \frac{8k}{2} = 4k \quad \text{(which is an integer)} \]
\[ \frac{x}{4} = \frac{8k}{4} = 2k \quad \text{(which is an integer)} \]

3. **Checking Divisibility by 3**:
The problem states that \( x \) is not divisible by 3. Therefore, we need to check the expression \( \frac{x}{6} \):
\[ \frac{x}{6} = \frac{8k}{6} = \frac{4k}{3} \]
For \( \frac{4k}{3} \) to be an integer, \( k \) must be divisible by 3. However, since \( x \) is not divisible by 3, \( k \) cannot be chosen such that \( 4k \) is divisible by 3. Hence, \( \frac{4k}{3} \) is not guaranteed to be an integer.

4. **Conclusion**:
Since \( x \) is divisible by 8 but not by 3, \( \frac{x}{6} \) cannot be an integer. Thus, the answer is:
\[ \text{The expression that cannot be an integer is } \frac{x}{6}. \]

### Final Answer:
The option that cannot be an integer is \( \frac{x}{6} \).

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