Starting at noon, João makes, every 80 minutes, a mark at the position of the hour hand of his clock.

a) After how much time will it no longer be necessary to make new marks on the clock?

b) What is the sum of the interior angles of the polygon formed by the marks?
Solution

a) The hour hand will complete one full revolution after $12 \cdot 60=720$ minutes, and during this time no mark will be repeated. Since 720 is a multiple of 80, during this period exactly $\frac{12 \cdot 60}{80}=9$ marks are made on the clock, and in addition, both hands return to their initial positions. Therefore, as the next marks will be repeated, the desired time is 720 minutes.

b) The sum of the interior angles of a 9-sided polygon is $180^{\circ} \cdot(9-2)=1260^{\circ}$.