(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Fix an integer [itex]p>1[/itex]. Prove that [itex]\mathbb Z_{p}[/itex] has exactly [itex]p[/itex] elements.

2. Relevant equations

Define the relation [itex]\equiv[/itex] on [itex]\mathbb Z[/itex] by setting [itex]a\equiv b[/itex] iff [itex]p|b-a[/itex]. (We have shown [itex]\equiv[/itex] to be an equivalence relation on [itex]\mathbb Z[/itex]). Let [itex]\mathbb Z_{p}=\{[a]:a\in\mathbb Z\}[/itex], where [itex][a]=\{b\in\mathbb Z:a\equiv b\}[/itex].

3. The attempt at a solution

I've unsuccessfully tried to show that if it had less than [itex]p[/itex] elements, then the union of the equivalence classes would not 'fill' [itex]\mathbb Z[/itex], and so [itex]\equiv[/itex] would not partition [itex]\mathbb Z[/itex], and so it would not be an equivalence relation: a contradiction. But I just can't get there, and I feel that that is much more of a difficult way to go about, and that there is a much easier route, any ideas?

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# Homework Help: Help with Z_p

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