Units group of Z modulo

[Bachelier]

I came across this question. Is the group (ℤ/34/ℤ)^x cyclic?
We haven't discussed the theorem in class that any units group of Z modulo n is iff n = 1, 2, 4, p^k and 2p^k (where p is an odd prime). But thanks to Deveno I know about it. So in this case, p =17 works, so the group is cyclic?

but is there a different way to show it besides using brute force .

 

[Bachelier]

BTW, p doesn't have to be equal to 4k+3 for this to be true, correct?

 

[Bachelier]

I guess I can show that ℤ/34/ℤ^x is ≈ to ℤ/17/ℤ^x and by the field's finite subgroup theorem, it is a cyclic.