Pretty easy question probably: How do you prove that $\phi(k)=(n-k)\mod{n}$ satisfies the homomorphism property for the binary structures $\langle\mathbb{Z}_n,+\rangle$ and $\langle\mathbb{Z}_n,+\rangle$?
Proof of Homomorphism Property for $(n - k)\mod n$
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abstract-algebra
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0@Mike105 : Your question was answered by Douglas S. Stones, but notice that you can find the answer by clicking on "edit" on your question and seeing what appears there. – 2012-09-20
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You need to show that for all $a$ and $b$, we have $\phi(a+b)=\phi(a)+\phi(b)$. So, to use number-theoretic terminology, you need to show that if $x\equiv -a\pmod{n}$ and $y\equiv -b\pmod{n}$, then $x+y\equiv -(a+b)\pmod{n}$.
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0Yes, working backwards worked for me. Thanks. – 2012-09-20