Find all primitive roots modulo $18.$

Question

• Find all primitive roots modulo $18.$

My Attempt. I need to show $ord_{18} (a)=\phi(18)=5.$

$5^4=13 \bmod 18$, I couldn't find primitve root mod $18$, can you help? Can you add an answer? Thanks...

Answer 1

The multiplicative group of invertible elements modulo $18$, $(\mathbb Z/18\mathbb Z)^\ast,$

has $\phi(18)=\phi(2)\phi(9)=\phi(9)=6$ elements: $\{1,5,7,11,13,17\}$.

The order of $1$ is $1$, the order of $17$ is $2$, the order of $7$ and its inverse $13$ is $3$,

and the order of $5$ and its inverse $11$ is $6$.

So the primitive roots mod $18$ are $5$ and $11$.