Problem 75

To solve the problem, let's denote the measure of each exterior angle of the polygon as $x$ degrees. Given that the measure of each interior angle is $5$ times the measure of its adjacent exterior angle, we can express the measure of each interior angle as $5x$ degrees.

Since the interior and exterior angles are supplementary (they add up to $180^{\circ}$), we can set up the equation:
$$x + 5x = 180^{\circ}$$

Simplifying this equation involves combining like terms:
$$6x = 180^{\circ}$$

To find the value of $x$, we divide both sides of the equation by $6$:
$$x = \frac{180^{\circ}}{6} = 30^{\circ}$$

Now that we know each exterior angle measures $30^{\circ}$, we can find the number of sides of the polygon. The sum of all exterior angles of a polygon is always $360^{\circ}$. Therefore, to find the number of sides ($n$), we divide the total measure of all exterior angles by the measure of one exterior angle:
$$n = \frac{360^{\circ}}{30^{\circ}} = 12$$

Thus, the value of $n$, which represents the number of sides of the polygon, is $\boxed{12}$.