$${3^{2022}}$$

$$ = {({3^2})^{1011}}$$

$$ = {(9)^{1011}}$$

$$ = {(10 - 1)^{1011}}$$

$$ = {}^{1011}{C_0}{(10)^{1011}} + \,\,.....\,\, + \,\,{}^{1011}{C_{1010}}\,.\,{(10)^1} - {}^{1011}{C_{1011}}$$

$$ = 10\left[ {{}^{1011}{C_0}{{(10)}^{1010}} + \,\,......\,\, + \,\,{}^{1011}{C_{1010}}} \right] - 1$$

$$ = 10\,K - 1$$

[As $$10\left[ {{}^{1011}{C_0}\,.\,{{(10)}^{1010}} + \,\,......\,\, + \,\,{}^{1011}{C_{1010}}} \right]$$ is multiple of 10]

$$ = 10K + 5 - 5 - 1$$

$$ = 10K - 5 + 5 - 1$$

$$ = 5(2K - 1) + 4$$

$$\therefore$$ Unit digit = 4 when divided by 5.