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)

Answer 1

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

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

Hence, the first term is $(160-17{)}^{\circ }={143}^{\circ },$
which is divisible by $1,11,13,143.$
$\therefore$ Number of divisors $=4$