In a regular polygon with $n$ sides, each interior angle can be calculated using the formula:
$$ ext{Interior Angle} =\frac{(n - 2) imes 180}{n}$$
Given that the sum of the interior angles of this polygon is equal to the sum of the interior angles of a triangle multiplied by $k$, where $k$ is an integer. If the sum of the interior angles of the triangle is $180$ degrees, then for which of the following values of $n$ is the polygon regular?