A number theory problem by Hritesh Mourya

$\begin{aligned} 11^{25} & \equiv 11^{10\times 2}11^5 \text{ (mod 100)} \\ & \equiv (10+1)^{10\times 2}11^5 \text{ (mod 100)} \\ & \equiv (10^{10} + 10\times 10^9+...+10\times 10 +1)^2 11^5 \text{ (mod 100)} \\ & \equiv 1\times 11^5 \text{ (mod 100)}\\ & \equiv 1\times 11^5 \text{ (mod 100)} \\ & \equiv 10^5 + 5 \times 10^4 + 10 \times 10^3 + 10 \times 10^2 + 5 \times 10 + 1 \text{ (mod 100)} \\ & \equiv \boxed{51} \text{ (mod 100)} \end{aligned}$