(a) Show that if $\displaystyle p$ is prime, all non-zero elements of $\displaystyle \mathbb{Z}_{p}$ have a multiplicative inverse.

(b) Deduce that $\displaystyle \mathbb{Z}^*_{p}$, the set of non-zero elements of $\displaystyle \mathbb{Z}_{p}$, is a group under multiplication mod $\displaystyle p$.

(c) Show that if $\displaystyle n$ is not prime, then $\displaystyle \mathbb{Z}^*_{n}$ is not a multiplicative group.

(d) Is $\displaystyle \mathbb{Z}^*_{17}$ cyclic? (prime order )

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