# Exterior angle of a regular polygon measures 10alpha degrees.Then, prove that alpha in ZZ and there are precisely 7 such alpha#?

Jul 18, 2018

We know, the magnitude of each external angle of a regular polygon of $n$ sides is related as follows.

Each external angle ($\theta$) $= {360}^{\circ} / n$,where n must be an positive integer $\ge 3$.

Now it is given $\theta = 10 \alpha$ degree

So $10 \alpha = {360}^{\circ} / n$

$\implies \alpha = \frac{36}{n}$

This shows that possible positive integral values of $\alpha$ are 12,9,6,4,3,2,1.

Hence $\alpha \in \mathbb{Z}$ takes precisely 7 values.