Download 2) Angles in a Star Polygon 3) Factors and Perfect

Extra Credit Problems Quarter IV
1) A rectangular courtyard can be covered by 1440 square tiles. If each tile were 2 inches
longer and wider, then it could instead be covered with 1210 tiles. Find the dimensions of
the courtyard.
2)
Angles in a Star Polygon
It is well known that the sum of the (interior) angles of a polygon can be found by the formula S =
180·(N-2) where S the sum of the angles, and N is the number of sides of the polygon.
A. Find the sum of the star point angles of this
7-gon, which is created by connecting every third point on the circle.
B. Find the sum of the star point angles of this
7-gon, which is created by connecting every second point on the circle.
C. Would your answers to the above two questions be different if the 7-gon were not regular (i.e., the
points were not evenly spaced on the circle)? Explain why.
D. Find the sum of the star point angles of a 17-gon, which is created by connecting every eighth point
on the circle.
E. Find the sum of the star point angles of a 17-gon, which is created by connecting every third point on
the circle.
F. Determine a formula (or method) by which you can calculate the sum of the angles of any star
polygon, where N is the number of points of the star, and every X number of points along the circle is
connected in order to create the star polygon.
3)
Factors and Perfect Numbers
A. How can we determine how many factors a given number has? 14 has four factors, 60
has twelve factors, and 48 has ten factors. But how can we determine the number of
factors a very large number has? For example, it turns out that the number
1,103,350,248,000 has 3360 factors. (How did I figure that out?) How many factors
does 9,489,150,000 have?
B. Crazy Factoring! Factor x12  1, and then list all of its binomial factors. (Note: a3 + b3
factors to (a+b)(a2ab+b2)) What laws can you find in all of this?
Now use these laws to help you factor x15  1.
C. Determine the prime factorization of 260  1.
D. Perfect Numbers. A perfect number is a number that is the sum of its factors (not
including the number itself). For example, 28 is a perfect number because its factors (1,
2, 4, 7, 14) add to 28. Perfect numbers were of great interest to the Greeks, but they only
knew the first four of them: 6, 28, 496, 8128. In spite of efforts, the fifth perfect number
remained undiscovered for over 1500 years. Determine the fifth perfect number.