4. The regular octagon $A B C D E F G H$ is inscribed in a circle. Points $P$ and $Q$ are on the circle, with $P$ between $C$ and $D$, such that $A P Q$ is an equilateral triangle. It is possible to inscribe a regular $n$-sided polygon, one of whose sides is $P D$, in the circle. What is the value of $n$ ?
SOlUTION
Given that $A B C D E F G H$ is a regular octagon, and letting its centre be $O$, we find that $\angle A O D=3 \times 360^{\circ} \div 8=135^{\circ}$.

Similarly, given that $A P Q$ is an equilateral triangle, we also have $\angle A O P=360^{\circ} \div 3=120^{\circ}$.

Therefore the angle $P O D=135^{\circ}-120^{\circ}=15^{\circ}$. Now, since 15 is factor of 360 , it is true that $P D$ is a side of a regular polygon inscribed in the circle. The number of sides, $n$, of this polygon is $360 \div 15=24$.