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I can't see why this group $\mathbb{Z} [1/p] / \mathbb{Z}$ is divisible. It is $p$-divisible, but how is it divisible by any integer $n$ when the denominator is only powers of $p$?

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Given $\frac a{p^k}+\Bbb Z$ and $n\in\Bbb N$ with $p\nmid n$ find $r$ such that $n\mid a+rp^k$ (possible because $p$ is invertible $\bmod n$), say $a+rp^k=nb$. Then $n\cdot (\frac b{p^k}+\Bbb Z)=\frac a{p^k}+\Bbb Z$.

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