Remainder Properties #1

$\begin{aligned} 9^{505} \times 3^{1011} + 1 & \equiv 9^{505} \times 3^{2\times 505+1} + 1 \pmod 5 \\ & \equiv 9^{505} \times 9^{505} \times 3 + 1 \pmod 5 \\ & \equiv 9^{1010} \times 3 + 1 \pmod 5 \\ & \equiv (10-1)^{1010} \times 3 + 1 \pmod 5 \\ & \equiv (-1)^{1010} \times 3 + 1 \pmod 5 \\ & \equiv 3 + 1 \pmod 5 \\ & \equiv \boxed 4 \pmod 5 \end{aligned}$

I used logic. The mod 5 will cycle around after 4 answers (without the plus 1, it's never going to be 0 so 5 – 1 = 4) so I changed the numbers into base 3 and modded the exponent with 4.

{ 1 + 9^505 × 3^1011 } mod 5
= { 1 + (3^2)^505 × 3^1011 } mod 5
= { 1 + 3^1010 × 3^1011 } mod 5
= { 1 + 3^(1010 + 1011) } mod 5
= { 1 + 3^2021 } mod 5
= { 1 + 3^[ 2021 mod 4 ] } mod 5
= { 1 + 3^[ 1 mod 4 ] } mod 5
= { 1 + 3^1 } mod 5
= { 1 + 3 } mod 5
= 4