Last two digits.

Relevant wiki: Euler's Theorem

$\begin{aligned} 2017^{2017} & \equiv 17^{2017} \text{ (mod 100)} \\ & \equiv 17^{\color{#3D99F6} 2017 \text{ mod }\phi (100)} \text{ (mod 100)} & \small \color{#3D99F6} \text{Since }\gcd(17, 100) = 1 \text{, Euler's theorem applies.} \\ & \equiv 17^{\color{#3D99F6} 2017 \text{ mod }40} \text{ (mod 100)} & \small \color{#3D99F6} \text{Euler's totient function }\phi (100) = 40 \\ & \equiv 17^{\color{#3D99F6}17} \text{ (mod 100)} \\ & \equiv 17\left(17^2\right)^8 \text{ (mod 100)} \\ & \equiv 17\left(289\right)^8 \text{ (mod 100)} \\ & \equiv 17\left(300-11\right)^8 \text{ (mod 100)} \\ & \equiv 17\times 11^8 \text{ (mod 100)} \\ & \equiv 17(10+1)^8 \text{ (mod 100)} \\ & \equiv 17(8\times 10 +1) \text{ (mod 100)} \\ & \equiv (10+7)(80 +1) \text{ (mod 100)} \\ & \equiv (560+10+7) \text{ (mod 100)} \\ & \equiv \boxed{77} \text{ (mod 100)} \end{aligned}$