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## Homework Statement

I just finished proving that ##\overline{a} \in \mathbb{Z}/n\mathbb{Z}## is nilpotent if and only if every prime divisor of ##n## is a prime divisor ##a##, and this lead me to wonder whether the cyclic subgroup ##\langle \overline{p_1 p_2 ... p_k} \rangle## be consist of all nilpotent elements, where ##p_1,...p_k## appear in the prime factorization of ##n##. This seems to be the case when ##n = 72##.

## Homework Equations

## The Attempt at a Solution

It seems that this would be an immediate corollary of the theorem to which I alluded. Is this right? Every element of ##\langle \overline{p_1 p_2 ... p_k} \rangle## is of the form ##\overline{x(p_1 p_2 ... p_k)}##, and it seems that every prime divisor of ##n## would be a divisor of ##x(p_1 p_2 ... p_k)##. Does this sound correct?