Basic concept but still difficult #4

$\phi (10^{ 5001 })={ 10 }^{ 5001 }\cdot (1-\frac { 1 }{ 2 } )\cdot (1-\frac { 1 }{ 5 } )\\ \Rightarrow \quad \phi (10^{ 5001 })={ 10 }^{ 5000 }\cdot 10 \cdot \frac { 1 }{ 2 } \cdot \frac { 4 }{ 5 } \\ \Rightarrow \quad \phi (10^{ 5001 })={ 10 }^{ 5000 }\cdot 4\\ \Rightarrow \quad 1000\quad |\quad \phi (10^{ 5001})\\ \Rightarrow \quad \phi (10^{ 5001 })\quad \equiv \quad 0\quad mod\quad 1000$