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For a fixed $n\in \mathbb{N}$, prove that

$\cos(\frac{jr\pi}{n})\neq 1$ if and only if $\gcd(j,2n)=1$, where $1\leq j,r\leq (n-1)$.

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1 Answer 1

$cos(\frac{jr\pi}{n}) = 1$ if and only if $\frac{jr\pi}{n} = 2k\pi$ for all $k \in \mathbb{Z}$. In particular, take k=1, So $\frac{jr}{n}=2$ i.e. $jr=2n$ $\Rightarrow$ j divides 2n. So, if j>1 $gcd(j,2n)\neq 1$. if j is 1 it is easy to prove.

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Thanks a lot.................... – user47929 Nov 3 '12 at 19:22

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