For any ordinal \alpha < \omega^N, there are unique n<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.