Find the Automotphism group of direct product of Z(mod m) & Z(mod n). Find the Automotphism group of direct product of $\mathbb{Z}(\operatorname{mod} m)$ and $\mathbb{Z}(\operatorname{mod} n)$. 
We know Automorphism of direct product of $\mathbb{Z}(\operatorname{mod} p)$ and $\mathbb{Z}(\operatorname{mod} p)$ is isomorphic to $\operatorname{GL}_2(\mathbb{Z}(\operatorname{mod} p))$. Then how to generalize it?
 A: There is a classical result by Shoda:
@article{shoda28,
author =       "Kenjiro Shoda",
title =        "{\"Uber die Automorphismen einer endlichen Abelschen Gruppe}",
journal =      "Mathematische Annalen",
volume =       100,
pages =        "674--686",
year =         1928
}

that describes the automorphism group of an abelian group. The paper is eminently readable (well, if you can read German...).
In short, his result says the following: First split up into powers of primes. Different Sylow subgroups are characteristic and the Automorphism group is simply the direct product of the automorphism groups of the Sylow subgroups. So without loss of generality, assume that $m=p^a$ and $n=p^b$ with $b\le a$ and generators $x$ (order $m$) and $y$ (order $n$).
If $a=b$ the automorphisms then can be described by an invertible $2\times 2$ matrix over $Z_{p^a}$.
If $a<b$, then $x$ needs to be mapped to $x^ey^f$ with $e$ coprime to $p$, and $y$ needs to be mapped to $y^kx^l$ with $k$ coprime to $p$, and $l$ a multiple of $p^{b-a}$. 
It is not hard to see that these conditions are not just necessary (to preserve order), but also that each these ''matrices'' $e,f;k,l$ gives an automorphism.
