$L_1 = \{ s \in (0+1)^* \mid d(s) \mod \ 5=2 \}$ is regular
Having $2$ as final state out of $\{0,1,2,3,4\}$ states
As given in example in posted link [in same DFA , final state will be $2$ instead of $0$ ]
Similarly, $L_2 = \{ s \in (0+1)^* \mid d(s) \mod \ 7 \neq 4 \}$ is also regular
Having states $\{0,1,2,3,4,5,6\}$ and all are final state except $4$
$L_1 \cap L_2$ is $L$ (given problem ) is also regular
As regular languages are closed under intersection. D is correct option.
Reference: https://gateoverflow.in/1695/gate1998_4