A number theory problem by Chaitanya Sriram

Relevant wiki: Euler's Theorem

$\begin{aligned} 19^{92} & \equiv 19^{92 \bmod \phi(92)} \text{ (mod 92)} & \small \color{#3D99F6} \text{Since }\gcd(19, 92) = 1\text{, Euler's theorem applies.} \\ & \equiv 19^{92 \bmod 44} \text{ (mod 92)} & \small \color{#3D99F6} \text{Euler's totient function }\phi(92) = 44 \\ & \equiv 19^4 \equiv (19^2)^2 \text{ (mod 92)} \\ & \equiv 361^2 \equiv (368-7)^2 \text{ (mod 92)} & \small \color{#3D99F6} \text{Note that }92 \times 4 = 368 \\ & \equiv (-7)^2 \equiv \boxed{49} \text{ (mod 92)} \end{aligned}$

From Euler's theorem we have $19^{44} = 1 mod 92$ so $19^{88}=1 mod 92$ .

Now we need to see the remainder of $19^{4}$ . $19^{2} = 361 = 85 = -7 mod 92$ so $19^{92} = -7^{2} = \boxed { 49} mod 92$