The units digit in the integral part of $(\sqrt{15}+\sqrt{220})^{19}(1+(\sqrt{15}+\sqrt{220})^{63})$ is:
Step-by-Step Solution
Key Concept: Let $A=\sqrt{15}+\sqrt{220}$, $B=\sqrt{15}-\sqrt{220}$ (conjugate, $|B|<1$). $A+B=2\sqrt{15}$ and $AB=15-220=-205$. $A^{19}+B^{19}=I_1+1$ (integer). $A^{82}+B^{82}=I_2+1$ (integer). So $I+I_1=10\lambda$ and $I_1+1=10\lambda_1$.
Units digit $=\mathbf{9}$.
Correct Answer: 9