ordinal, uniqueness question

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

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

I do not see how to use the hint to prove this. Would the $n$ 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.