$$\begin{aligned} &\begin{aligned} & 64^{64} \Rightarrow(63+1)^{64}=63 \lambda+1 \\ & 64^{664^{64}} \Rightarrow(63+1)^{64^{64}} \\ & 63 \lambda_1+1 \end{aligned}\\ &\text { Required remainder when divided } 7 \text { is } 1 \text {. } \end{aligned}$$