Problem 24 · 2015 Math Kangaroo
Hard
Number Theory
divisibility
How many regular n-sided shapes are there whose angles (in degrees) are whole numbers?
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Answer: C — 22
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Hint 1 of 2
A regular n-gon's interior angle is 180 − 360/n.
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Hint 2 of 2
It is a whole number exactly when n divides 360.
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Approach: count divisors of 360 that are at least 3
- The interior angle 180 − 360/n is a whole number exactly when n divides 360.
- 360 = 2³·3²·5 has 4×3×2 = 24 divisors; a polygon needs n ≥ 3, so drop n = 1 and 2.
- That leaves 22 valid values of n (C).
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