The remainder when 7 2022 + 3 2022 is divided by 5 is

Using the binomial expansion in the given expression we get,

$7^{2022}+3^{2022}={\left(50-1\right)}^{1011}+{\left(10-1\right)}^{1011}$

$={}^{1011}C_0{\left(50\right)}^{1011}{\left(-1\right)}^0+{}^{1011}C_1{\left(50\right)}^{1010}{\left(-1\right)}^1+....{}^{1011}C_{1011}{\left(50\right)}^0{\left(-1\right)}^{1011}$$+{}^{1011}C_0{\left(10\right)}^{1011}{\left(-1\right)}^0+{}^{1011}C_1{\left(10\right)}^{1010}{\left(-1\right)}^1+....{}^{1011}C_{1011}{\left(10\right)}^0{\left(-1\right)}^{1011}$

$=5k-2$

Using binomial expansion we get the remainder as $-2+5=3$