Is it possible to have a regular polygon with each interior angle equal to `105^@?`

Answer

To determine if it's possible to have a regular polygon with each interior angle equal to \(105^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Formula for Interior Angles**:
The formula for the interior angle \(i\) of a regular polygon with \(n\) sides is given by:
\[ i = \frac{(n - 2) \times 180^\circ}{n} \]

2. **Set Up the Equation**:
We need to check if \(i = 105^\circ\). Therefore, we set up the equation:
\[ \frac{(n - 2) \times 180^\circ}{n} = 105^\circ \]

3. **Cross Multiply**:
To eliminate the fraction, we can cross-multiply:
\[ (n - 2) \times 180^\circ = 105^\circ \times n \]

4. **Distribute and Rearrange**:
Expanding the left side gives us:
\[ 180n - 360 = 105n \]
Now, rearranging the equation to isolate \(n\):
\[ 180n - 105n = 360 \]

5. **Combine Like Terms**:
Simplifying the left side:
\[ 75n = 360 \]

6. **Solve for \(n\)**:
Dividing both sides by 75 gives:
\[ n = \frac{360}{75} = 4.8 \]

7. **Determine the Nature of \(n\)**:
Since \(n = 4.8\) is not a natural number (whole number), it indicates that a regular polygon with each interior angle of \(105^\circ\) is not possible.

### Conclusion:
Thus, it is not possible to have a regular polygon with each interior angle equal to \(105^\circ\).

---