Grogg draws an equiangular polygon with $g$ sides, and Winnie draws an equiangular polygon with $w$ sides. If the interior angle of Winnie's polygon is equal to five times the exterior angle of Grogg's polygon, find the largest possible value of $w$.

Answer 1

In Grogg's polygon, each interior angle has measure $\frac{180(g-2)}{g}$ degrees and so the measure of each exterior angle is $\frac{180}{g}-\frac{360}{g}= \frac{-180}{g}$ degrees.

Similarly, in Winnie's polygon, each interior angle has measure $\frac{180(w-2)}{w}$ degrees. Since the exterior angle in Grogg's polygon is five times this measure, we have $$5\cdot \frac{180(w-2)}{w}=\frac{-180}{g}$$Clearing out the fractions gives $5(w-2)=-g.$

Winnie's number of sides, $w,$ is an integer. This equation tells us that $g$ is divisible by 5. Recall that $g$ is also Grogg's number of sides, so the largest value Winnie's number of sides can be is $\boxed{5}.$