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MATH / Find all positive integer values of n that satisfy the equation

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Find all positive integer values of nn that satisfy the equation \[ \cos \Bigl( \frac{\pi}{n} \Bigr) \cos \Bigl( \frac{2\pi}{n} \Bigr) \cos \Bigl( \frac{4\pi}{n} \Bigr) \cos \Bigl( \frac{8\pi}{n} \Bigr) \cos \Bigl( \frac{16\pi}{n} \Bigr) = \frac{1}{32}. \]Enter all the solutions, separated by commas.
Plain-text mathematical notation (without MathML)
Find all positive integer values of n that satisfy the equation
\[
  \cos \Bigl( \frac{\pi}{n} \Bigr) \cos \Bigl( \frac{2\pi}{n} \Bigr)   
     \cos \Bigl( \frac{4\pi}{n} \Bigr) \cos \Bigl( \frac{8\pi}{n} \Bigr)
     \cos \Bigl( \frac{16\pi}{n} \Bigr)
    = \frac{1}{32}.
\]Enter all the solutions, separated by commas.
Original LaTeX notation
Find all positive integer values of $n$ that satisfy the equation
\[
  \cos \Bigl( \frac{\pi}{n} \Bigr) \cos \Bigl( \frac{2\pi}{n} \Bigr)   
     \cos \Bigl( \frac{4\pi}{n} \Bigr) \cos \Bigl( \frac{8\pi}{n} \Bigr)
     \cos \Bigl( \frac{16\pi}{n} \Bigr)
    = \frac{1}{32}.
\]Enter all the solutions, separated by commas.

level

Level 5

type

Precalculus

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Source and history

task_id
928d6017-f955-5856-a907-2365471acc63
task_key
precalculus--test--928d6017-f955-5856-a907-2365471acc63
task_revision_id
1
config / split
precalculus / test

Official source

initial import