 Solution

Removing an $e$-hook removes one cell of every residue modulo $e$. If $\lambda$ and $\mu$ have the same $e$-core and the same size, part a(i) shows that they have the same $e$-weight. Their residue multisets are therefore obtained from the common core by adjoining the same number of complete residue sets $\{0,1,\dots ,e-1\}$. Thus their $e$-contents are equal.