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Let $p$ and $q$ be two different primes. Let $f$ be a homomorphism of $\mathbb{Z}/pq\mathbb{Z}$ onto $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/q\mathbb{Z}$. I want to show that in this case $f$ is bijective and find the multiplicative inverse of $f$.
I found that the only such $f$ are $f(x)=\big(x(p+1),x(q+1)\big)$ by noting that the image of $1$ is $(0,0)$ and $f(x+y)=f(x)+f(y)$ for $x,y\in\mathbb{Z}/pq\mathbb{Z}$.
This part is straightforward.
For the second part I need to find the multiplicative inverse of $f$ if $f$ is bijective. If $f$ is surjective I’m done since $\mathbb{Z}/pq\mathbb{Z}$ has at least two elements so we can find it by taking $x$ to be an element in $f(\mathbb{Z}/pq\mathbb{Z})$. In the case that $f$ is only injective (i.e. only $f$ is bijective), we need to find an element $y$ such that $f(y)=f(x)$ and $f(y)
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