Let, ‘ $\in$’ be nth root of unity and z be a complex number such that  $|z-{\in }^k|\le 1$$\forall k=0,1,\mathrm{....},n-1$ . Then  $\left|z\right|$ can be___________

$(z-{\in }^k)(\overline{z}-{\in }^{\overline{k}})\le 1$

$z\overline{z}-z\overline{{\in }^k}-\overline{Z}{\in }^k+{\in }^k\overline{{\in }^k}\le 1$

$\Rightarrow {\left|z\right|}^2\le z\overline{{\in }^k}+\overline{z}{\in }^k$

Now, sum these relations  $\Rightarrow {\left|z\right|}^2\times n\le z\left(\sum _{k=0}^{n-1}{\in }^k\right)+\overline{z}\left(\sum _{k=0}^{n-1}{\in }^k\right)$

$\Rightarrow {\left|z\right|}^2\cdot n\le 0\Rightarrow \left|z\right|=0$