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Prove that $$\mathbb Z_m/\langle \overline{n}\rangle \cong\mathbb Z_{\text{gcd}(m,n)}.$$ for any $m,n\in \mathbb{N}$.

For this, I know I must show that in $\mathbb Z_m$, $\langle \overline{n} \rangle$ and $\langle \text{gcd}(m,n)\rangle$ are the same ideal, but I'm not sure how I could possibly show that. Any suggestions?

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Hint: Consider the projection $\Bbb Z_m \to \Bbb Z_{gcd(m,n)}$ and the first isomorphism theorem.

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  • $\begingroup$ So I want to show that it maps one-to-one and onto, correct? $\endgroup$ – Brian Nov 11 '13 at 2:13
  • $\begingroup$ Yes for the 'onto', no for the 'one-to-one', note that if $d = gcd(m,n)$, then $d$ projects to $0$. $\endgroup$ – Vinicius M. Nov 11 '13 at 2:17
  • $\begingroup$ How do I show that it is onto? I know that it maps onto S provided its image is all of S, but I don't know how to prove this for a problem like this. $\endgroup$ – Brian Nov 11 '13 at 3:57

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