1. ## Z/nZ - Exercise

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 $\displaystyle \bar{11}^{31}$ = $\displaystyle \bar{14}$.

2. ## Re: Z/nZ - Exercise

(d)
we need to solve the equation

$\displaystyle 11x \equiv 1 \bmod 17$

Show that

$\displaystyle 11*14=154\equiv 1 \bmod 17$

since we also have from part (c) that $\displaystyle 11*11^{31}\equiv 1 \bmod 17$

it follows that $\displaystyle 11^{31}\equiv 14 \bmod 17$