Let $n$ a positive integer and $k \in \mathbb{Z}^n$ such that for all integer $a \geq 2$ and $h \in \mathbb{Z}^n$ we have $k \neq ah$. Here $\cdot$ is the scalar product.

Is it true that $\{x \in \mathbb{Z}^n : k \cdot x = 0\} \cong \mathbb{Z}^n$ (group isomorphism)?

The answer is yes for $n=1,2$ but the general case seems difficult to me.

Thanks for any suggestions.

Let $n$ a positive integer and $k \in \mathbb{Z}^n$ such that for all integer $a \geq 2$ and $h \in \mathbb{Z}^n$ we have $k \neq ah$. Here $\cdot$ is the scalar product.

Is it true that $\{x \in \mathbb{Z}^n : k \cdot x = 0\} \cong \mathbb{Z}^{n-1}$ (group isomorphism)?

The answer is yes for $n=1,2$ but the general case seems difficult to me.

Thanks for any suggestions.