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Problem 94 (When is $x \pmod d$ a homomorphism from $\mathbb{Z}_n$ to $\mathbb{Z}_d$)

Let $G=\mathbb{Z}_{12}$ and $H=\mathbb{Z}_5$. Let $f:G\to H$ be the map defined by $f(x)=x\mod 5$.

  1. Is the map $f:G\to H$ a homomorphism? Either prove that it is, or produce a counter example.
  2. Let $n$ be an integer and suppose that the positive integer $d<n$ is not a divisor of $n$. Is the map $f:\mathbb{Z}_n\to \mathbb{Z}_d$ defined by $f(x)=x\mod d$ a homomorphism?

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