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

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 $U_{209}$.

• For a less obvious solution, solve $x\equiv 10\pmod{11}$, $x\equiv -10\pmod{19}$ using the Chinese Remainder Theorem. – André Nicolas Jul 28 '14 at 3:31

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 $x^4\equiv 100\pmod{209}$.

• Thank you!! That helps so much! I understand it now. – B. L. Miller Jul 28 '14 at 3:29
• @B.L.Miller: You could post your solution as an answer and accept it so we have such. This is encouraged. – Ross Millikan Jul 28 '14 at 3:48
• Thank you, I just did! I'm new to this website, I don't quite understand how to use it yet. Thank you for the tip! – B. L. Miller Jul 28 '14 at 3:57

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 $x=155$.

• Note that you only need to use Euclid's algorithm to get the first two: note that 142=209-67 and 199=209-10. So these are equivalent modulo 209 to -67 and -10, and therefore square to 100. Same idea for the fourth roots. – Semiclassical Jul 28 '14 at 4:00
• Hopefully you don't need Euclid's algorithm for the first one. – Dylan Stephano-Shachter Jul 28 '14 at 13:51