(Solved): 0. (i) Prove that \( \phi:(\mathbb{Z},+) \rightarrow Z_{n}=\langle x\rangle \) with \( \phi(m)=x^{m ...
0. (i) Prove that \( \phi:(\mathbb{Z},+) \rightarrow Z_{n}=\langle x\rangle \) with \( \phi(m)=x^{m} \) is a homomorphism; (ii) Find the kernel \( K=\phi^{-1}\left(1_{Z_{n}}\right) \) and the image \( \phi(\mathbb{Z}) \); (iii) Use the results from (i) and (ii) to conclude that \( \mathbb{Z} /(n \mathbb{Z}) \cong Z_{n} \).