Question

Which one of the followings is constructible by using compass and straightedge? Why

A. regular 7-gon

B. regular 37-gon

C. regular 85-gon

D. regular 97-gon

Answer 1

Restating the Gauss-Wantzel theorem:

A regular n-gon is constructible with straightedge and compass if and only if n = 2^kp₁p₂...p_t where k and t are non-negative integers, and the p_i's (when t > 0) are distinct Fermat primes.

The five known Fermat primes are:

F₀ = 3, F₁ = 5, F₂ = 17, F₃ = 257, and F₄ = 65537

Since there are 31 combinations of anywhere from one to five Fermat primes, there are 31 known constructible polygons with an odd number of sides.

The next twenty-eight Fermat numbers, F₅ through F₃₂, are known to be composite.

Thus a regular n-gon is constructible if

n = 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96, 102, 120, 128, 136, 160, 170, 192, 204, 240, 255, 256, 257, 272, 320, 340, 384, 408, 480, 510, 512, 514, 544, 640, 680, 768, 771, 816, 960, 1020, 1024, 1028, 1088, 1280, 1285, 1360, 1536, 1542, 1632, 1920, 2040, 2048, ...

while a regular n-gon is not constructible with compass and straightedge if

n = 7, 9, 11, 13, 14, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 97, 98, 99, 100, 101, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, ...

So C) Regular 85-gon is constructible by using compass and straightedge

Here 85=2^0*5*17(k=0,p1=5,p2=17).