If `(361)/(181 xx 181)` is an integer, then find the highest power of 19 by which it is divisible.

Options

A. 2
B. 3
C. 5
D. 6

Answer

To solve the problem, we need to determine the highest power of 19 that divides the expression \(\frac{361}{181 \times 181}\).

### Step-by-Step Solution:

1. **Identify the components of the expression**:
We have the expression \(\frac{361}{181 \times 181}\).

2. **Factor 361**:
We recognize that \(361\) can be expressed as \(19^2\) because \(19 \times 19 = 361\).

\[ 361 = 19^2 \]

3. **Factor \(181\)**:
Next, we need to check if \(181\) is divisible by \(19\). We can do this by dividing \(181\) by \(19\):

\[ 181 \div 19 \approx 9.5263 \]

Since \(181\) is not divisible by \(19\), we conclude that \(181\) does not contribute any factors of \(19\).

4. **Rewrite the expression**:
Now we can rewrite the original expression using our findings:

\[ \frac{361}{181 \times 181} = \frac{19^2}{181 \times 181} \]

5. **Determine the highest power of 19**:
In the numerator, we have \(19^2\). In the denominator, \(181\) does not contain any factors of \(19\). Therefore, the highest power of \(19\) in the entire expression is simply the power from the numerator:

\[ \text{Highest power of } 19 = 2 \]

6. **Conclusion**:
Thus, the highest power of \(19\) by which \(\frac{361}{181 \times 181}\) is divisible is \(2\).

### Final Answer:
The highest power of \(19\) by which the expression is divisible is \(2\).