The difference between an exterior angle of (n - 1) sided regular polygon and an exterior angle of (n + 2) sided regular polygon is $6^{\circ}$ . Find the value of n.

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Solution

The correct option is A 13
An exterior angle of $(n - 1)$ sided regular polygon = $\frac{360^{o}}{(n - 1)}$
An exterior angle of $(n + 2)$ sided regular polygon = $\frac{360^{o}}{(n + 2)}$
Their difference is $6^{o}$
So,
$\frac{360^{o}}{(n - 1)} - \frac{360^{o}}{(n + 2)} = 6^{o}$
$=> 60 \frac{(n + 2) - (n - 1)}{(n - 1) (n + 2)} = 1$
$=> (n - 1) (n + 2) = 180$
$=> n^{2} + n - 2 = 180$
$=> n^{2} + n - 182 = 0$
$=> n^{2} + 14 n - 13 n - 182 = 0$
$=> (n - 13) (n - 14) = 182$
$=> n = 13, - 14$
Since $n$ should be a positive number. So, $n = 13$

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