To prove that two cyclic subgroups H and K within a cyclic group G are the same when their orders are equal

Question

I see in Fraleigh's abstract algebra book, there is a theorem stating that two cyclic subgroups $H$ and $K$ of a cyclic group $G$, denoted as <$a^s$> and <$a^t$>,respectively, are the same, iff their orders are the same. I don't understand why the cyclic subgroup of $Z_n$ contains all the integers m less than n such that $gcd(m,n)=d$. Or is there any better and brief proof for this?

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