$\text{Future of }\red{2020}?$

$2027$ is prime number

By Fermat's Little Theorem , for any positive integer $a$ and prime $p$

$a^{p-1} \equiv 1 \bmod p$

$2020^{2027 - 1} \equiv 1 \bmod 2027$

$2020^{2026} \equiv 1 \bmod 2027 \ldots (1)$

$2020 \equiv -7 \bmod 2027$

$2020^2 \equiv (-7)(-7) \bmod 2027$

$2020^2 \equiv 49 \bmod 2027 \ldots (2)$

$2020^{2026} × 2020^2 \equiv 49\bmod 2027 \ldots \text{(by (1) and (2))}$

$\color{#3D99F6}{\boxed{2020^{2028} \equiv 49 \bmod 2027}}$

$\begin{aligned} 2020^{2028} & \equiv (2027-7)^{2028} \text{ (mod 2027)} \\ & \equiv (-7)^{2028} \text{ (mod 2027)} & \small \blue{\text{Since }\gcd(7,2027) =1 \text{, Euler's theorem applies.}} \\ & \equiv 7^{2028 \bmod \blue{\phi(2027)}} \text{ (mod 2027)} & \small \blue{\text{And the Euler's totient function }\phi(2027) - 2027-1,} \\ & \equiv 7^{2028 \bmod \blue{2026}} \text{ (mod 2027)} & \small \blue{\text{since 2027 is a prime.}} \\ & \equiv 7^2 \equiv \boxed{49} \text{ (mod 2027)} \end{aligned}$

References:

• Euler's theorem
• Euler's totient function