A number theory problem by Rahil Sehgal

Relevant wiki: Euler's Theorem

$\begin{aligned} 3^{100000} & \equiv 3^{\color{#3D99F6}100000 \text{ mod }\phi (53)} \text{ (mod 53)} & \small \color{#3D99F6} \text{Since } \gcd(3,53) = 1 \text{, Euler's theorem applies.} \\ & \equiv 3^{\color{#3D99F6}100000 \text{ mod }52} \text{ (mod 53)} & \small \color{#3D99F6} \text{Euler's totient function }\phi(53) = 52 \\ & \equiv 3^{10(104-4)^2 \text{ mod }52} \text{ (mod 53)} \\ & \equiv 3^{160 \text{ mod }52} \text{ (mod 53)} \\ & \equiv 3^4 \text{ (mod 53)} \\ & \equiv 81 \text{ (mod 53)} \\ & \equiv \boxed{28} \text{ (mod 53)} \end{aligned}$