All about that Base!

$\begin{aligned} 7^{999} & \equiv 7^{\color{#3D99F6}{999 \text{ mod } \phi(1000)}} \text{ (mod 1000)} & \small \color{#3D99F6}{\text{Since }\gcd (7,1000) = 1 \text{, Euler's theorem applies}} \\ & \equiv 7^{\color{#3D99F6}{999 \text{ mod } 400}} \text{ (mod 1000)} & \small \color{#3D99F6}{\text{Euler's totient function }\phi (1000) = 400} \\ & \equiv 7^{199} \text{ (mod 1000)} \\ & \equiv 7^{199} \text{ (mod 1000)} \\ & \equiv 7^{4\times 49+3} \text{ (mod 1000)} & \small \color{#3D99F6}{\text{Note that } 7^4 = 2401, \ 7^3 = 343} \\ & \equiv 2401^{49} \cdot 343 \text{ (mod 1000)} \\ & \equiv 401^{49} \cdot 343 \text{ (mod 1000)} \\ & \equiv (400+1)^{5\times 9 +4} \cdot 343 \text{ (mod 1000)} & \small \color{#3D99F6}{\text{Note that } (400+1)^5 \equiv 1 \text{ (mod 1000)}} \\ & \equiv (400+1)^4 \cdot 343 \text{ (mod 1000)} & \small \color{#3D99F6}{\text{Note that } (400+1)^4 \equiv 601 \text{ (mod 1000)}} \\ & \equiv 601 \cdot 343 \text{ (mod 1000)} \\ & = \boxed{143} \text{ (mod 1000)} \end{aligned}$

${ 7 }^{ 4n }\equiv { (1+400n) }^{ n }\equiv 1+400n\quad (mod\quad 1000)$ .