Let $\alpha=\frac{(4 !) !}{(4 !)^{3 !}}$ and $\beta=\frac{(5 !) !}{(5 !)^{4 !}}$. Then :

Question

$\alpha \in \mathbf{N}$ and $\beta \in \mathbf{N}$

$\alpha \in \mathbf{N}$ and $\beta \notin \mathbf{N}$

$\alpha \notin \mathbf{N}$ and $\beta \in \mathbf{N}$

$\alpha \notin \mathbf{N}$ and $\beta \notin \mathbf{N}$

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