I was reading an article about the number of group morphisms from $\mathbb{Z}_n$ to $\mathbb{Z}_m$ for $m,n>0$ and it turns out it's the same number of cosets $[x]\in\mathbb{Z}_m$ that satisfy $nx\equiv 0 \mod m$; it also said that the number of solutions is $ \gcd(m,n)$ but it didn't explain why. Any explanation to this?
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The solutions are $x=\dfrac{km}{\gcd(m,n)}$ for $k=1$ to $\gcd(m,n)$ – J. W. Tanner Mar 07 '24 at 17:57