$$ \begin{aligned} & 7^{2022}+3^{2022} \\\\ & =\left(7^2\right)^{1011}+\left(3^2\right)^{1011} \\\\ &=(50-1)^{1011}+(10-1)^{1011} \\\\ &= (50^{1011}-1011.50^{1010}+\ldots-1) \\\\ & + (10^{1011}-1011.10^{1010}+\ldots . .-1) \\\\ &= 5 m-1+5 n-1=5(m+n)-2 \\\\ &= 5(m+n)-5+3=5(m+n-1)+3 \\\\ &= 5 k+3 \\\\ & \therefore \text { Remainder }=3 \end{aligned} $$