Hi All,

I was hoping that someone could check my answers/working for the following questions.

Question 1)

Find the order of 2 & 3 in$\displaystyle \mathbb{Z} / 13 \mathbb{Z}$. Which one's the primitive root?

(I don't know the latex to show all my working so I wont show all of it.)

$\displaystyle \varphi (13) \ = \ 12 $

$\displaystyle 2 $ has order $\displaystyle 12 \ = \ \varphi(13) $ and is therefore a primitive root.

3 has order 3 and isn't a primitive root.

Question 2) Using the primitive root, solve $\displaystyle x^{11} \ = \ 4(mod \ 13) $.

My solution.......

$\displaystyle 4 \ \equiv \ 2^2 \ (mod \ 13) $

Since 2 is a primitive root, $\displaystyle x \ $ is some power of $\displaystyle 2 \ mod \ 13 $

Now, letting $\displaystyle x = 2^y $, then

$\displaystyle x^{11} \ \equiv \ 4 \ (mod \ 13)$ becomes $\displaystyle (2^y)^{11} = 2^{11y} \ \equiv \ 2^2 \ (mod \ 13)$

Then since 2 has order $\displaystyle \varphi (13) = 12$,

$\displaystyle 11y \equiv \ 2(mod \ 12) $.

Multiplying both sides by 4 gives

$\displaystyle y = 8 \ mod \ 12$ which we substitute back to get

$\displaystyle x \ \equiv \ 2^8 \ \equiv \ 4 \ mod \ 12$

Is this correct? Any help would be greatly appreciated.

Thanks