Hello,

I don't know how to solve the last question, thanks for your help

(a) Justify $\displaystyle \bar{11}$ ∈ (Z/17Z)*

11 is co-prime with 17 hence 11 ∈ (Z/17Z)*

(b) What is $\displaystyle \bar{11}^{16}$ in Z/17Z ? .

According to Fermat's little theorem

$\displaystyle \bar{11}^{16} \equiv 1 $ [17]

(c) Show that $\displaystyle \bar{11}^{31}$ is a solution for the equation 11x ≡ 1 mod 17.

$\displaystyle 11^{31} = 11^{16} \times 11^{15} = 1 \times 11^{15}$

so $\displaystyle 11 \times 11^{15} \equiv 1 mod 17$

$\displaystyle 11^{16} \equiv 1 [17] $

$\displaystyle 1 \equiv 1[17] $

(d) Deduce that $\displaystyle \bar{11}^{31}$ = $\displaystyle \bar{14}$.