What is the remainder when \(2^{202}+202\) is divided by \(2^{101}+2^{51}+1\)?
Solution
Answer: \(201\)
Complete the square, then apply the difference of squares:
\[\begin{aligned}2^{202}+202&=(2^{101}+1)^2-2^{102}+201\&=(2^{101}-2^{51}+1)(2^{101}+2^{51}+1)+201.\end{aligned}\]
The product is divisible by the given divisor, leaving remainder \(201\).