# If an element $cinmathbb{U}_m$ has order $o$, then $mathbb{U}_m$ has elements of every order less than $o$

I’m trying to prove that, if some $$cinmathbb{U}_m$$ has order $$o$$ (i.e. $$c^o = 1$$), $$mathbb{U}_m$$ contains elements of every order less than $$o$$. It seems to me that perhaps this has something to do with $$mathbb{U}_m$$ being a cyclic group, however I’m not sure where to begin with this.

Edit: $$mathbb{U}_m$$ is the set of units in $$mathbb{Z}_m$$

Mathematics Asked by K_M on November 12, 2021

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