Question 174
A complex number $z$ is called a root of unity if $z^k = 1$, for some positive integer $k$.
Given a positive integer $n$, let $S_n$ be the set of all roots of unity of the form
$$
\frac{a+bi}{\sqrt{n}}
$$
where $i = \sqrt{-1}$ and $a$ and $b$ are positive integers.
(i) Prove that $S_n$ has less than $\sqrt{n}$ distinct elements.
(ii) Prove that $S_5$ is empty.
Given a positive integer $n$, let $S_n$ be the set of all roots of unity of the form
$$
\frac{a+bi}{\sqrt{n}}
$$
where $i = \sqrt{-1}$ and $a$ and $b$ are positive integers.
(i) Prove that $S_n$ has less than $\sqrt{n}$ distinct elements.
(ii) Prove that $S_5$ is empty.