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What is the remainder when $2^{2023}+2023$ is divided by $2^{20}+1$? (Adapted From 2020 AMC 10B Problems, Question \#22) 
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Let $x=2^{20}$.  We are now looking for the remainder of $\frac{8x^{101}+2023}{x+1}$. By Polynomial Remainder Theorem, the remainder is $8\times (-1)^{101} + 2023 = 2015$. 