What are the values of $w$ such that the corresponding $\text{Keccak-}f[25w]$ function is invertible (bijective)?
$\operatorname{Keccak-}\!f[25w]$ is defined, by the Keccak specification, only when $w = 2^\ell$ for some $\ell$, and in all such cases it is a permutation.  The choices 1, 2, and 4 for $w$ are not multiples of 8.  No multiple of 5 is a power of 2 and consequently $w$ is never a multiple of 5.