(a) Find the multiplicative inverse of $14$ in $\mathbb{Z}_{15}$.


(b) Find the multiplicative inverse of $31$ in $\mathbb{Z}_{32}$.


(c) In general, what is the multiplicative inverse of $( n - 1 )$ in $\mathbb{Z}_{n}$?


(d) (Optional: see your instructor) Prove that your answer for (c) is correct.