Is it true that $\mathbb{R}^{d}/n\mathbb{Z}^{d} = n(\mathbb{S}^{1})^{d}$?

Question

I know that $\mathbb{R}/\mathbb{Z} = \mathbb{S}^{1}$ is the unit sphere and also $\mathbb{R}^{d}/\mathbb{Z}^{d} \cong (\mathbb{S}^{1})^{d}$. But given a positive integer $n$, is it true that $\mathbb{R}/n\mathbb{Z} = n\mathbb{S}^{1}$ and $\mathbb{R}^{d}/n\mathbb{Z}^{d} = n(\mathbb{S}^{1})^{d}$? In loose terms, should $\mathbb{R}^{d}/n\mathbb{Z}^{d}$ be treated like $[0,n)^{d}$? If so, how to address this problem to prove it?