This problem is No. 2.21 from the book "Introduction to the theory of groups," by Rotman.
Let $G = \langle a \rangle$ be a cyclic group of order $st$, where $\gcd (s,t) = 1$. Show that there are unique $b, c \in G$ with $b$ of order $s$ and $c$ of order $t$, and $a=bc$.
I have tried proving there's always a unique integer $y$ where $ys\equiv 0\pmod {st}$ and $(y+1)t\equiv 0\pmod {st}$, and then take $a^{y+1}$ and $a^{-y}$ as my generators after proving non of them generates a smaller group than $|s|$ and $|t|$ respectively, but failed to do so.
Hints\solutions are welcome.
Since $gcd(s,t)=1$, there exist integers $n$ ad $m$ such that $ns+mt=1$. In fact, these integers can be chosen so that $\lvert n \rvert <t$ and $\lvert m \rvert <s$. Now, choose $b=a^{mt}$ and $c=a^{ns}$.