If $z_1$ and $z_2$ be the n th root of unity which subtend right angled at the origin. Then, n must be of the form 

The nth roots of unity is given by

$\cos \left(\frac{2r\pi }{n}\right)+i\sin \left(\frac{2r\pi }{n}\right)=e^{2r\pi i/n}$

where $r=0,1,2,\dots .(n-1)$

So, let  $z_1=e^{2\pi r_{1}i/n}$ and $z_2=e^{2\pi r_{2}i/n}$, where $0\le r_1,r_2<n$,

$r_1\ne r_2$

It is given that the line segment joining the points having affixes z₁ and z₂, subtends a right angled at the origin. Therefore,

$\begin{array}{l}\arg \left(\frac{z_1}{z_2}\right)=\pm \frac{\pi }{2}\\ \Rightarrow \frac{2\pi r_1}{n}-\frac{2\pi r_2}{n}=\pm \frac{\pi }{2}\\ \Rightarrow n=\pm 4\left(r_1-r_2\right)\end{array}$

$\therefore n=4k,$ where $k=\pm \left(r_1-r_2\right)$