Is the Units Group of Z Modulo 34 Cyclic?

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The units group of ℤ modulo 34, denoted as (ℤ/34ℤ)x, is cyclic. This conclusion is supported by the theorem stating that the units group of ℤ modulo n is cyclic if n equals 1, 2, 4, pk, or 2pk, where p is an odd prime. In this case, since 34 can be expressed as 2 × 17, where 17 is an odd prime, the group is indeed cyclic. Additionally, it can be shown that (ℤ/34ℤ)x is isomorphic to (ℤ/17ℤ)x, confirming its cyclic nature through the finite subgroup theorem.

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Bachelier
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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, pk and 2pk (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 .
 
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BTW, p doesn't have to be equal to 4k+3 for this to be true, correct?
 
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I guess I can show that ℤ/34/ℤx is ≈ to ℤ/17/ℤx and by the field's finite subgroup theorem, it is a cyclic.
 

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