Problem:
The measures (in degrees) of the interior angles of a convex hexagon form an arithmetic sequence of positive integers. Let m∘ be the measure of the largest interior angle of the hexagon. The largest possible value of m∘ is
Answer Choices:
A. 165∘
B. 167∘
C. 170∘
D. 175∘
E. 179∘
Solution:
Let d be the common difference of the arithmetic sequence. The sum of the interior angles of the hexagon,
6m−15d=m+(m−d)+(m−2d)+⋯+(m−5d)=(6−2)180=720
shows that 6m=15d+720=5(3d+144), so m is divisible by 5. Because the hexagon is convex, m≤175. Because 65+87+109+131+153+175=720, there is such a hexagon and 175∘ is the answer