imc

Completing the square, then difference of squares: $$\begin{array}{rl}{2}^{202}+202& =(2^{101}{)}^2+2\cdot 2^{101}+1-2\cdot 2^{101}+201\\ & =(2^{101}+1{)}^2-2^{102}+201\\ & =(2^{101}-2^{51}+1)(2^{101}+2^{51}+1)+201.\end{array}$$ Thus, we see that the remainder is $\boxed{201}$