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List all elements of order $4$ in $\mathbb{Z}_8=\mathbb{Z}/8\mathbb{Z}$.

Also list all the elements of order $6$ in $\mathbb{Z}_{72}=\mathbb{Z}/72\mathbb{Z}$.

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What do you know in general about the subgroups of a cyclic group? – Andrea Mori Apr 2 '13 at 13:58

Hint: List all elements of $\Bbb Z_8$, then write up the order of each: $$\Bbb Z_8=\{0,1,2,3,4,-3,-2,-1\}\,,$$ as $-x$ represents the same equivalence class as $8-x$ modulo $8$. I guess, we are talking about additive order, that is, the question is for each $x\in\Bbb Z_8$, what is the least $n>0$, such that $n\cdot x=0$ (modulo $8$, of course).

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+1 for my friend Berci and for theoretical solid approach. – Babak S. Apr 2 '13 at 15:07

I did your problem by using GAP, however, it is strongly suggested that you do the Group theory exercises by hand. enter image description here

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