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**Capillarian** Let $\displaystyle (x_1,\ldots,x_r)$ be a non-zero element of $\displaystyle \mathbb{Z}^r$, and let h be the highest common factor of $\displaystyle x_1,\ldots,x_r$. Show that there is an isomorphism $\displaystyle \mathbb{Z}^r \to \mathbb{Z}^r$ taking $\displaystyle (x_1,\ldots,x_r)$ to (1, 0, 0,..., 0) if and only if h = 1.

Deduce that $\displaystyle \mathbb{Z}^r / \langle (x_1,\ldots,x_r)\rangle \cong \mathbb{Z}^{r-1} \oplus (\mathbb{Z}/h\mathbb{Z})$.

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The first part isn't a problem. But the deduction is something I'm having trouble with... help?