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Question

The degree measures of the angles in a convex 18-sided polygon form an increasing arithmetic sequence with integer values. If x is the degree measure of the smallest angle, then the number of divisors of x is 
(correct answer + 3, wrong answer 0)


Solution

The average angle in an 18-sided polygon is 160.
In an arithmetic sequence, the average is the same as the median, so the middle two terms of the sequence average to 160.
Thus for some positive (the sequence is increasing) integer d, the middle two terms are (160d) and (160+d).

Since the step is 2d, the last term of the sequence is (160+17d) which must be less than 180 as the polygon is convex.
160+17d<180
17d<20
So the only suitable positive integer d is 1.

Hence, the first term is (16017)=143,
which is divisible by 1,11,13,143.
Number of divisors =4

Mathematics

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