# Primitive roots, specifically of 18

this problem is annoying. ive found that the primitive roots of 9 are 2 and 5. since 2|18 it cant be a root. i know via some theorems in my book that if 5 is a primitive root of 3, then its a primitive root of 3^k, and also of 2*3^k.

sorry about not using latex, shouldnt need it for this one though, it wont be too messy im sure.

so the second one, i know i can find by brute force, but id like something more elegant. i know there are only phi of phi of 18, or 2, primitive roots.

i know also by table lookup that the second one of 18 is 11.

do i need to start with the gcd(a,18) list and just eliminate ones until i cant eliminate 11? that doesnt seem right, i remember seeing something in class about this, but the teacher said not to record it in notes because it was in the book, but the book only identifies the one, because i think it is always the smallest. or do i just take 2 mod 9 congruent to 11 mod 9, and thats it? sorry if i just missed something plain as day.