Exponents

Note that 2^12 ≡ 96 (mod 100) ≡ −4 (mod 100). Thus, 2^100 ≡ (2^12)^8(2^4) (mod 100) ≡ (−4)^8 2^4 (mod 100) ≡ 2^20 (mod 100) ≡ (−4)2^8 (mod 100) ≡ 76 (mod 100)

As 2 and 100 are not coprime integers, we cannot apply Euler's theorem straightaway. Let us consider $2^{100} \text{ mod } 4$ and $2^{100} \text{ mod } 25$ separately, as $100 = 4 \times 25$ .

$\begin{aligned} 2^{100} & \equiv 0 \text{ (mod 4)} \\ 2^{100} & \equiv 2^{\color{#3D99F6}{100 \text{ mod }\phi(25)}} \text{ (mod 25)} & \small \color{#3D99F6}{\text{Since }\gcd(2,25)=1 \text{, Euler's theorem applies.}} \\ 2^{100} & \equiv 2^{\color{#3D99F6}{100 \text{ mod } 20}} \text{ (mod 25)} & \small \color{#3D99F6}{\text{Euler totient function }\phi(25) = 20} \\ 2^{100} & \equiv 2^{\color{#3D99F6}{0}} \equiv 1 \text{ (mod 25)} \\ \implies 25n+1 & \equiv 0 \text{ (mod 4)} \\ n + 1 & \equiv 0 \text{ (mod 4)} \\ n & \equiv -1 \text{ (mod 4)} \\ n & \equiv 3 \text{ (mod 4)} \\ \implies 2^{100} & \equiv 25n+1 \text{ (mod 100)} \\ & \equiv \boxed{76} \text{ (mod 100)} \end{aligned}$

$2^{100} = 1267650600228229401496703205376$