For any ordinal $\displaystyle \alpha < \omega^N$, there are unique $\displaystyle n<N$, $\displaystyle x< \omega$, $\displaystyle \beta < \omega^n$ such that $\displaystyle \alpha=\omega^n \cdot x + \beta$.

Hint: Choose $\displaystyle n$ largest such that $\displaystyle \omega^n \leq \alpha$, and $\displaystyle x$ largest such that $\displaystyle \omega^n \cdot x \leq \alpha$.

I do not see how to use the hint to prove this. Would the $\displaystyle n$ found in the hint be unique? Then I was struggling showing that the other two were unique. I would appreciate some hints on this. Thanks.