# Finding a primitive root modulo $11^2$

Find a primitive root modulo each of the following moduli:

a) $11^2$

My TA said he is not going to go over this so do not worry about it. He said you can try this if you want but he would not go over this.

Can someone please show me how to solve this problem? It seems like something good to know.

• Primitive roots are quite common (if $r$ is a primitive root modulo $m$, so is any power of $r$ that is relatively prime to $\phi(m)$, i.e., there are $\phi(\phi(m))$ of them), so a few random tries should net you one. – vonbrand Apr 30 '13 at 19:44

In general, if $a$ is a primitive root modulo $p$ then either $a$ or $a+p$ is a primitive root modulo $p^2$.
So find a primitive root, $a$, modulo $11$, then check $a$ and $a+p$ modulo $11^2$.