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## Main Question or Discussion Point

Hello. So today in class, we talked a bit about [itex]<g>[/itex] as the set of all integer powers of [itex]g[/itex]. Made enough sense. Then we did some examples in [itex]\mathbb{Z} / 7\mathbb{Z}[/itex], and I got a bit lost. I think this is more due to the fact that perhaps I don't quite grasp [itex]\bmod{n}[/itex] as an equivalence class. So I know that the elements of [itex]\mathbb{Z} / 7\mathbb{Z}[/itex] are [itex]\{ [0], [1], [2], [3], [4], [5], [6] \}[/itex], but we were looking for [itex]<g>[/itex] where [itex]g = [5][/itex]. The idea was we were showing that [itex]\mathbb{Z} / 7\mathbb{Z}[/itex] is cyclic. Since we were "brute-forcing" it so to speak, we wrote down elements in a certain order, which I have written down as [itex]\{ [0], [5], [3], [1], [6], [4], [2] \}[/itex] and then it cycled back to [itex][0][/itex].

So I suppose I'm having trouble doing computations in [itex]\mathbb{Z} / 7\mathbb{Z}[/itex], or [itex]\mathbb{Z} / n\mathbb{Z}[/itex] in general. I mean, I know that if [itex]a \equiv b \bmod{n}[/itex] then [itex]n \mid a-b[/itex], but in this case I'm not sure where to take it. Is it:

[itex]5^0 = 1 \equiv[/itex] something [itex]\bmod{7}[/itex], then [itex]5^1 = 5 \equiv[/itex] something [itex]\bmod{7}[/itex] etc?

Sorry if my question(s) is a bit hard to read; perhaps I didn't explain it as best I could. But any help would be appreciated.

So I suppose I'm having trouble doing computations in [itex]\mathbb{Z} / 7\mathbb{Z}[/itex], or [itex]\mathbb{Z} / n\mathbb{Z}[/itex] in general. I mean, I know that if [itex]a \equiv b \bmod{n}[/itex] then [itex]n \mid a-b[/itex], but in this case I'm not sure where to take it. Is it:

[itex]5^0 = 1 \equiv[/itex] something [itex]\bmod{7}[/itex], then [itex]5^1 = 5 \equiv[/itex] something [itex]\bmod{7}[/itex] etc?

Sorry if my question(s) is a bit hard to read; perhaps I didn't explain it as best I could. But any help would be appreciated.