The least integer greater than ` log_(2) 15* log_(1//6 2* log_(3) 1//6` is _______.

Explanation

To solve the problem, we need to evaluate the expression \( \log_2 15 \cdot \log_{1/6} 2 \cdot \log_{3} \frac{1}{6} \) and find the least integer greater than this value.

### Step-by-step Solution:

1. **Rewrite the logarithms using properties**:
We know that \( \log_{a} b = \frac{\log_{c} b}{\log_{c} a} \). Using this property, we can rewrite the logarithms in terms of a common base (let's use base 10 for simplicity).

\[ \log_2 15 = \frac{\log_{10} 15}{\log_{10} 2} \]
\[ \log_{1/6} 2 = \frac{\log_{10} 2}{\log_{10} (1/6)} = \frac{\log_{10} 2}{-\log_{10} 6} \]
\[ \log_{3} \frac{1}{6} = \frac{\log_{10} (1/6)}{\log_{10} 3} = \frac{-\log_{10} 6}{\log_{10} 3} \]

2. **Substituting back into the expression**:
Now substituting these back into the original expression:

\[ \log_2 15 \cdot \log_{1/6} 2 \cdot \log_{3} \frac{1}{6} = \left(\frac{\log_{10} 15}{\log_{10} 2}\right) \cdot \left(\frac{\log_{10} 2}{-\log_{10} 6}\right) \cdot \left(\frac{-\log_{10} 6}{\log_{10} 3}\right) \]

3. **Simplifying the expression**:
Notice that \( \log_{10} 2 \) and \( -\log_{10} 6 \) cancel out:

\[ = \frac{\log_{10} 15}{\log_{10} 3} \]

4. **Evaluating \( \log_{10} 15 \)**:
We can express \( 15 \) as \( 3 \cdot 5 \):

\[ \log_{10} 15 = \log_{10} (3 \cdot 5) = \log_{10} 3 + \log_{10} 5 \]

5. **Final expression**:
Thus, we have:

\[ = \frac{\log_{10} 3 + \log_{10} 5}{\log_{10} 3} \]

This simplifies to:

\[ = 1 + \frac{\log_{10} 5}{\log_{10} 3} \]

6. **Estimating the value**:
We know that \( \log_{10} 3 \approx 0.477 \) and \( \log_{10} 5 \approx 0.699 \). Therefore:

\[ \frac{\log_{10} 5}{\log_{10} 3} \approx \frac{0.699}{0.477} \approx 1.465 \]

So:

\[ 1 + \frac{\log_{10} 5}{\log_{10} 3} \approx 1 + 1.465 \approx 2.465 \]

7. **Finding the least integer greater than the value**:
The least integer greater than \( 2.465 \) is \( 3 \).

### Final Answer:
The least integer greater than \( \log_2 15 \cdot \log_{1/6} 2 \cdot \log_{3} \frac{1}{6} \) is **3**.