$${(2021)^{2022}} + {(2022)^{2021}}$$

$$ = {(7k - 2)^{2022}} + {(7{k_1} - 1)^{2021}}$$

$$ = {\left[ {{{(7k - 2)}^3}} \right]^{674}} + {(7{k_1})^{2021}} - 2021{(7{k_1})^{2020}}\, + \,....\, - 1$$

$$ = {(7{k_2} - 1)^{674}} + (7m - 1)$$

$$ = (7n + 1) + (7m - 1) = 7(m + n)$$ (multiple of 7)

$$\therefore$$ Remainder = 0