Which of the following statements are true? Justify your answers. (a) Every ordinal is of the form $\alpha+n$, where $\alpha$ is a limit ordinal and $n \in \omega$. (b) Every ordinal is of the form $\omega^{\alpha} \cdot m+n$, where $\alpha$ is an ordinal and $m, n \in \omega$. (c) If $\alpha=\omega \cdot \alpha$, then $\alpha=\omega^{\omega} \cdot \beta$ for some $\beta$. (d) If $\alpha=\omega^{\alpha}$, then $\alpha$ is uncountable. (e) If $\alpha>1$ and $\alpha=\alpha^{\omega}$, then $\alpha$ is uncountable.

[Standard laws of ordinal arithmetic may be assumed, but if you use the Division Algorithm you should prove it.]