Find the $4$ sq. roots of $100$ in $ U_{209}$. Identify which square root of $100$ is square.

Question

Find the $4$ sq. roots of $100$ in $U_{209}$. Identify which square root of $100$ is square.

(Not the $4^{th}$ root, but the $4$ square roots). I honestly don't even know what this question is asking... All I can say is $\sqrt{100}$ is $10$. Don't know what that has to do with $U_{209}$.

Answer 1

I think $\mathrm U_{209}$ means $(\mathbf Z/209\mathbf Z)^\times$. It then asks for the four $1\leq x\leq 208$ such that $x^2\equiv 100 \pmod{209}$. It will be useful to note that $209=11\times 19$.

The square square root problem should then be: find $1\leq x \leq 208$ such that $x^4\equiv 100\pmod{209}$.

Answer 2

I used Euclid's algorithm to get $x= 10$, $x=67$, $x= 142$, and $x= 199$.

I did the same for the square square root and got $x=54$, $x= 98$, $x=111$, and $x=155$.