1962 AHSME Problems/Problem 31
The ratio of the interior angles of two regular polygons with sides of unit length is . How many such pairs are there?
The formula for the measure of the interior angle of a regular polygon with -sides is . Letting our two polygons have side length and , we have that the ratio of the interior angles is . Cross multiplying both sides, we have . Using Simon's Favorite Factoring Trick, we have . Because and are both more than , we know that . Now, we just set these factors equal to the factors of 24. We can set to , , , or and to , , , or respectively to get the following pairs for : , , , and . However, we have to take out the solution with , because and are both more than , leaving us with as the correct answer.