The greatest integer less than or equal to the number `log_2(15)xx log_(1/6)2xx log_3(1/6)` is

Options

A. 4
B. 3
C. 2
D. 1

Explanation

To solve the problem, we need to evaluate the expression \( \log_2(15) \times \log_{1/6}(2) \times \log_3(1/6) \) and find the greatest integer less than or equal to this value.

### Step-by-Step Solution:

1. **Rewrite the logarithms using the change of base formula:**
\[ \log_2(15) = \frac{\log(15)}{\log(2)}, \quad \log_{1/6}(2) = \frac{\log(2)}{\log(1/6)}, \quad \log_3(1/6) = \frac{\log(1/6)}{\log(3)} \]

2. **Substituting these into the original expression:**
\[ \log_2(15) \times \log_{1/6}(2) \times \log_3(1/6) = \left(\frac{\log(15)}{\log(2)}\right) \times \left(\frac{\log(2)}{\log(1/6)}\right) \times \left(\frac{\log(1/6)}{\log(3)}\right) \]

3. **Notice that \(\log(2)\) and \(\log(1/6)\) cancel out:**
\[ = \frac{\log(15)}{\log(3)} \]

4. **Express \(\log(15)\) in terms of \(\log(3)\):**
\[ \log(15) = \log(3 \times 5) = \log(3) + \log(5) \]
Thus,
\[ \frac{\log(15)}{\log(3)} = \frac{\log(3) + \log(5)}{\log(3)} = 1 + \frac{\log(5)}{\log(3)} \]

5. **Now we need to evaluate \(1 + \frac{\log(5)}{\log(3)}\):**
- We know that \(\log(5) \approx 0.6990\) and \(\log(3) \approx 0.4771\).
- Therefore,
\[ \frac{\log(5)}{\log(3)} \approx \frac{0.6990}{0.4771} \approx 1.465 \]
Thus,
\[ 1 + \frac{\log(5)}{\log(3)} \approx 1 + 1.465 = 2.465 \]

6. **Finding the greatest integer less than or equal to \(2.465\):**
The greatest integer less than or equal to \(2.465\) is \(2\).

### Final Answer:
The greatest integer less than or equal to \( \log_2(15) \times \log_{1/6}(2) \times \log_3(1/6) \) is \( \boxed{2} \).