Number Theory

Relevant wiki: Euler's Theorem

Similar to @Ezra Manatad 's solution

$\begin{aligned} 331^{62} & \equiv 331^{\color{#3D99F6}62 \text{ mod }\lambda (2100)} \text{ (mod 2100)} & \small \color{#3D99F6} \text{Since }\gcd (331, 2100)=1 \text{, Euler's theorem applies.} \\ & \equiv 331^{\color{#3D99F6}62 \text{ mod }60} \text{ (mod 2100)} & \small \color{#3D99F6} \text{Carmichael's lambda function }\lambda(2100) = 60 \text{ (see note)} \\ & \equiv 331^2 \text{ (mod 2100)} \\ & \equiv 109561 \text{ (mod 2100)} \\ & \equiv \boxed{361} \text{ (mod 2100)} \end{aligned}$

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Note: Carmichael's lambda function

$\begin{aligned} \lambda (2100) & = \text{lcm}\left(\lambda(2^2), \lambda(3), \lambda(5^2), \lambda(7) \right) \\ & = \text{lcm} \left(2, 2, 20, 6 \right) \\ & = 60 \end{aligned}$

lambda(2100)=LCM(6,2,20)=60

331^62 (mod 2100) =(331²)(331^60) (mod 2100) =331² (mod 2100) =109561 (mod 2100) =361 (mod 2100)